diff --git a/ChangeLog.md b/ChangeLog.md
--- a/ChangeLog.md
+++ b/ChangeLog.md
@@ -1,4 +1,18 @@
-# Changelog for galois-field
+# Change log for galois-field
+
+## 0.2.0
+
+* Add `deg` for GaloisField
+* Add `order` for GaloisField
+* Add `pow` for GaloisField
+* Add `rnd` for GaloisField
+
+## 0.1.1
+
+* Add `Arbitrary` instances to PrimeField, PolynomialRing, and ExtensionField.
+* Add `Bits` instances to PrimeField.
+* Add `Pretty` instances to PrimeField, PolynomialRing, and ExtensionField.
+* Minor optimisations to multiplication and inversion with `INLINE`.
 
 ## 0.1.0
 
diff --git a/README.md b/README.md
--- a/README.md
+++ b/README.md
@@ -67,8 +67,8 @@
 The following data type declaration creates a splitting polynomial given an irreducible monic polynomial.
 ```haskell
 data P2
-instance IrreducibleMonic Fq P2
-  where split _ = x^2 + 1
+instance IrreducibleMonic Fq P2 where
+  split _ = x^2 + 1
 ```
 The following type declaration then creates an extension field with this splitting polynomial.
 ```haskell
diff --git a/benchmarks/Benchmarks.hs b/benchmarks/Benchmarks.hs
deleted file mode 100644
--- a/benchmarks/Benchmarks.hs
+++ /dev/null
@@ -1,184 +0,0 @@
-module Benchmarks where
-
-import Protolude
-
-import Criterion.Main
-
-import ExtensionField
-import PrimeField
-
-type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583
-
-fq :: Fq
-fq = 5216004179354450092383934373463611881445186046129513844852096383579774061693
-
-fq' :: Fq
-fq' = 10757805228921058098980668000791497318123219899766237205512608761387909753942
-
-data Pu
-instance IrreducibleMonic Fq Pu where split _ = x^2 + 1
-type Fq2 = ExtensionField Fq Pu
-
-fq2 :: Fq2
-fq2 = fromList
-  [ 19908898611787582971615951530393785823319364696376311494770162270472288380562
-  , 2444690988583914246674870181013910409542697083717824402984851238236041783759
-  ]
-
-fq2' :: Fq2
-fq2' = fromList
-  [ 176307305890807650390915550856467756101144733976249050387177647283239486934
-  , 9913547941088878400547309488585076816688958962210000330808066250849942240036
-  ]
-
-data Pv
-instance IrreducibleMonic Fq2 Pv where split _ = x^3 - (9 + t x)
-type Fq6 = ExtensionField Fq2 Pv
-
-fq6 :: Fq6
-fq6 = fromList
-  [ fromList
-    [ 8727269669017421992537561450387212506711577304101544328736696625792447584819
-    , 14548604791762199086915107662335514800873255588931510951007415299299859294564
-    ]
-  , fromList
-    [ 12226353852518517213098257637254082040554292743096797524265221809863992104040
-    , 12690801089710533803594523982915673248220237967492611523932652691226365708512
-    ]
-  , fromList
-    [ 18336930404004840796680535059992401039831316705513753839479258873269709495858
-    , 21634580953983557175729336703450663797341055784728343534694506874757389871868
-    ]
-  ]
-
-fq6' :: Fq6
-fq6' = fromList
-  [ fromList
-    [ 21427158918811764040959407626476119248515601360702754918240300689672054041331
-    , 12750457256357562507331331307761996193149796736574153338180573114576232473092
-    ]
-  , fromList
-    [ 19307896751125425658868292427117755307914453765471505616446813557567103424424
-    , 11511704315039881938763578963465960361806962511008317843374696569679546862720
-    ]
-  , fromList
-    [ 16856354813335682789816416666746807604324955216244680818919639213184967817815
-    , 10563739714379631354612735346769824530666877338817980746884577737330686430079
-    ]
-  ]
-
-data Pw
-instance IrreducibleMonic Fq6 Pw where split _ = x^2 - t x
-type Fq12 = ExtensionField Fq6 Pw
-
-fq12 :: Fq12
-fq12 = fromList
-  [ fromList
-    [ fromList
-      [ 4025484419428246835913352650763180341703148406593523188761836807196412398582
-      , 5087667423921547416057913184603782240965080921431854177822601074227980319916
-      ]
-    , fromList
-      [ 8868355606921194740459469119392835913522089996670570126495590065213716724895
-      , 12102922015173003259571598121107256676524158824223867520503152166796819430680
-      ]
-    , fromList
-      [ 92336131326695228787620679552727214674825150151172467042221065081506740785
-      , 5482141053831906120660063289735740072497978400199436576451083698548025220729
-      ]
-    ]
-  , fromList
-    [ fromList
-      [ 7642691434343136168639899684817459509291669149586986497725240920715691142493
-      , 1211355239100959901694672926661748059183573115580181831221700974591509515378
-      ]
-    , fromList
-      [ 20725578899076721876257429467489710434807801418821512117896292558010284413176
-      , 17642016461759614884877567642064231230128683506116557502360384546280794322728
-      ]
-    , fromList
-      [ 17449282511578147452934743657918270744212677919657988500433959352763226500950
-      , 1205855382909824928004884982625565310515751070464736233368671939944606335817
-      ]
-    ]
-  ]
-
-fq12' :: Fq12
-fq12' = fromList
-  [ fromList
-    [ fromList
-      [ 495492586688946756331205475947141303903957329539236899715542920513774223311
-      , 9283314577619389303419433707421707208215462819919253486023883680690371740600
-      ]
-    , fromList
-      [ 11142072730721162663710262820927009044232748085260948776285443777221023820448
-      , 1275691922864139043351956162286567343365697673070760209966772441869205291758
-      ]
-    , fromList
-      [ 20007029371545157738471875537558122753684185825574273033359718514421878893242
-      , 9839139739201376418106411333971304469387172772449235880774992683057627654905
-      ]
-    ]
-  , fromList
-    [ fromList
-      [ 9503058454919356208294350412959497499007919434690988218543143506584310390240
-      , 19236630380322614936323642336645412102299542253751028194541390082750834966816
-      ]
-    , fromList
-      [ 18019769232924676175188431592335242333439728011993142930089933693043738917983
-      , 11549213142100201239212924317641009159759841794532519457441596987622070613872
-      ]
-    , fromList
-      [ 9656683724785441232932664175488314398614795173462019188529258009817332577664
-      , 20666848762667934776817320505559846916719041700736383328805334359135638079015
-      ]
-    ]
-  ]
-
-benchmarks :: Benchmark
-benchmarks = bgroup "GaloisField"
-  [ bgroup "PrimeField"
-    [ bgroup "Fq"
-      [ bench "Addition"
-        $ whnf (uncurry (+)) (fq, fq')
-      , bench "Subtraction"
-        $ whnf (uncurry (-)) (fq, fq')
-      , bench "Multiplication"
-        $ whnf (uncurry (*)) (fq, fq')
-      , bench "Division"
-        $ whnf (uncurry (/)) (fq, fq')
-      ]
-    ]
-  , bgroup "ExtensionField"
-    [ bgroup "Fq2"
-      [ bench "Addition"
-        $ whnf (uncurry (+)) (fq2, fq2')
-      , bench "Subtraction"
-        $ whnf (uncurry (-)) (fq2, fq2')
-      , bench "Multiplication"
-        $ whnf (uncurry (*)) (fq2, fq2')
-      , bench "Division"
-        $ whnf (uncurry (/)) (fq2, fq2')
-      ]
-    , bgroup "Fq6"
-      [ bench "Addition"
-        $ whnf (uncurry (+)) (fq6, fq6')
-      , bench "Subtraction"
-        $ whnf (uncurry (-)) (fq6, fq6')
-      , bench "Multiplication"
-        $ whnf (uncurry (*)) (fq6, fq6')
-      , bench "Division"
-        $ whnf (uncurry (/)) (fq6, fq6')
-      ]
-    , bgroup "Fq12"
-      [ bench "Addition"
-        $ whnf (uncurry (+)) (fq12, fq12')
-      , bench "Subtraction"
-        $ whnf (uncurry (-)) (fq12, fq12')
-      , bench "Multiplication"
-        $ whnf (uncurry (*)) (fq12, fq12')
-      , bench "Division"
-        $ whnf (uncurry (/)) (fq12, fq12')
-      ]
-    ]
-  ]
diff --git a/benchmarks/Main.hs b/benchmarks/Main.hs
--- a/benchmarks/Main.hs
+++ b/benchmarks/Main.hs
@@ -3,8 +3,200 @@
 import Protolude
 
 import Criterion.Main
+import ExtensionField
+import PrimeField
 
-import Benchmarks
+type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583
 
+fq :: Fq
+fq = 5216004179354450092383934373463611881445186046129513844852096383579774061693
+
+fq' :: Fq
+fq' = 10757805228921058098980668000791497318123219899766237205512608761387909753942
+
+data Pu
+instance IrreducibleMonic Fq Pu where
+  split _ = x^2 + 1
+type Fq2 = ExtensionField Fq Pu
+
+fq2 :: Fq2
+fq2 = fromList
+  [ 19908898611787582971615951530393785823319364696376311494770162270472288380562
+  , 2444690988583914246674870181013910409542697083717824402984851238236041783759
+  ]
+
+fq2' :: Fq2
+fq2' = fromList
+  [ 176307305890807650390915550856467756101144733976249050387177647283239486934
+  , 9913547941088878400547309488585076816688958962210000330808066250849942240036
+  ]
+
+data Pv
+instance IrreducibleMonic Fq2 Pv where
+  split _ = x^3 - (9 + t x)
+type Fq6 = ExtensionField Fq2 Pv
+
+fq6 :: Fq6
+fq6 = fromList
+  [ fromList
+    [ 8727269669017421992537561450387212506711577304101544328736696625792447584819
+    , 14548604791762199086915107662335514800873255588931510951007415299299859294564
+    ]
+  , fromList
+    [ 12226353852518517213098257637254082040554292743096797524265221809863992104040
+    , 12690801089710533803594523982915673248220237967492611523932652691226365708512
+    ]
+  , fromList
+    [ 18336930404004840796680535059992401039831316705513753839479258873269709495858
+    , 21634580953983557175729336703450663797341055784728343534694506874757389871868
+    ]
+  ]
+
+fq6' :: Fq6
+fq6' = fromList
+  [ fromList
+    [ 21427158918811764040959407626476119248515601360702754918240300689672054041331
+    , 12750457256357562507331331307761996193149796736574153338180573114576232473092
+    ]
+  , fromList
+    [ 19307896751125425658868292427117755307914453765471505616446813557567103424424
+    , 11511704315039881938763578963465960361806962511008317843374696569679546862720
+    ]
+  , fromList
+    [ 16856354813335682789816416666746807604324955216244680818919639213184967817815
+    , 10563739714379631354612735346769824530666877338817980746884577737330686430079
+    ]
+  ]
+
+data Pw
+instance IrreducibleMonic Fq6 Pw where
+  split _ = x^2 - t x
+type Fq12 = ExtensionField Fq6 Pw
+
+fq12 :: Fq12
+fq12 = fromList
+  [ fromList
+    [ fromList
+      [ 4025484419428246835913352650763180341703148406593523188761836807196412398582
+      , 5087667423921547416057913184603782240965080921431854177822601074227980319916
+      ]
+    , fromList
+      [ 8868355606921194740459469119392835913522089996670570126495590065213716724895
+      , 12102922015173003259571598121107256676524158824223867520503152166796819430680
+      ]
+    , fromList
+      [ 92336131326695228787620679552727214674825150151172467042221065081506740785
+      , 5482141053831906120660063289735740072497978400199436576451083698548025220729
+      ]
+    ]
+  , fromList
+    [ fromList
+      [ 7642691434343136168639899684817459509291669149586986497725240920715691142493
+      , 1211355239100959901694672926661748059183573115580181831221700974591509515378
+      ]
+    , fromList
+      [ 20725578899076721876257429467489710434807801418821512117896292558010284413176
+      , 17642016461759614884877567642064231230128683506116557502360384546280794322728
+      ]
+    , fromList
+      [ 17449282511578147452934743657918270744212677919657988500433959352763226500950
+      , 1205855382909824928004884982625565310515751070464736233368671939944606335817
+      ]
+    ]
+  ]
+
+fq12' :: Fq12
+fq12' = fromList
+  [ fromList
+    [ fromList
+      [ 495492586688946756331205475947141303903957329539236899715542920513774223311
+      , 9283314577619389303419433707421707208215462819919253486023883680690371740600
+      ]
+    , fromList
+      [ 11142072730721162663710262820927009044232748085260948776285443777221023820448
+      , 1275691922864139043351956162286567343365697673070760209966772441869205291758
+      ]
+    , fromList
+      [ 20007029371545157738471875537558122753684185825574273033359718514421878893242
+      , 9839139739201376418106411333971304469387172772449235880774992683057627654905
+      ]
+    ]
+  , fromList
+    [ fromList
+      [ 9503058454919356208294350412959497499007919434690988218543143506584310390240
+      , 19236630380322614936323642336645412102299542253751028194541390082750834966816
+      ]
+    , fromList
+      [ 18019769232924676175188431592335242333439728011993142930089933693043738917983
+      , 11549213142100201239212924317641009159759841794532519457441596987622070613872
+      ]
+    , fromList
+      [ 9656683724785441232932664175488314398614795173462019188529258009817332577664
+      , 20666848762667934776817320505559846916719041700736383328805334359135638079015
+      ]
+    ]
+  ]
+
 main :: IO ()
-main = defaultMain [benchmarks]
+main = defaultMain
+  [ bgroup "PrimeField"
+    [ bgroup "Fq"
+      [ bench "Addition"
+        $ whnf (uncurry (+)) (fq, fq')
+      , bench "Multiplication"
+        $ whnf (uncurry (*)) (fq, fq')
+      , bench "Negation"
+        $ whnf negate fq
+      , bench "Subtraction"
+        $ whnf (uncurry (-)) (fq, fq')
+      , bench "Inversion"
+        $ whnf recip fq
+      , bench "Division"
+        $ whnf (uncurry (/)) (fq, fq')
+      ]
+    ]
+  , bgroup "ExtensionField"
+    [ bgroup "Fq2"
+      [ bench "Addition"
+        $ whnf (uncurry (+)) (fq2, fq2')
+      , bench "Multiplication"
+        $ whnf (uncurry (*)) (fq2, fq2')
+      , bench "Negation"
+        $ whnf negate fq2
+      , bench "Subtraction"
+        $ whnf (uncurry (-)) (fq2, fq2')
+      , bench "Inversion"
+        $ whnf recip fq2
+      , bench "Division"
+        $ whnf (uncurry (/)) (fq2, fq2')
+      ]
+    , bgroup "Fq6"
+      [ bench "Addition"
+        $ whnf (uncurry (+)) (fq6, fq6')
+      , bench "Multiplication"
+        $ whnf (uncurry (*)) (fq6, fq6')
+      , bench "Negation"
+        $ whnf negate fq6
+      , bench "Subtraction"
+        $ whnf (uncurry (-)) (fq6, fq6')
+      , bench "Inversion"
+        $ whnf recip fq6
+      , bench "Division"
+        $ whnf (uncurry (/)) (fq6, fq6')
+      ]
+    , bgroup "Fq12"
+      [ bench "Addition"
+        $ whnf (uncurry (+)) (fq12, fq12')
+      , bench "Multiplication"
+        $ whnf (uncurry (*)) (fq12, fq12')
+      , bench "Negation"
+        $ whnf negate fq12
+      , bench "Subtraction"
+        $ whnf (uncurry (-)) (fq12, fq12')
+      , bench "Inversion"
+        $ whnf recip fq12
+      , bench "Division"
+        $ whnf (uncurry (/)) (fq12, fq12')
+      ]
+    ]
+  ]
diff --git a/galois-field.cabal b/galois-field.cabal
--- a/galois-field.cabal
+++ b/galois-field.cabal
@@ -2,12 +2,12 @@
 --
 -- see: https://github.com/sol/hpack
 --
--- hash: 86cbd60c516785b91dbc82303018451a9a632edd09640a077a30faab5d3b4991
+-- hash: f66dcf977899a69f9f8785c75595dada58f68cbc1cb35faa4acb24b494fe26fe
 
 name:           galois-field
-version:        0.1.0
-synopsis:       Galois field
-description:    Finite field library
+version:        0.2.0
+synopsis:       Galois field library
+description:    Galois field library for cryptography research
 category:       Cryptography
 homepage:       https://github.com/adjoint-io/galois-field#readme
 bug-reports:    https://github.com/adjoint-io/galois-field/issues
@@ -24,16 +24,6 @@
   type: git
   location: https://github.com/adjoint-io/galois-field
 
-flag optimized
-  description: Perform compiler optimizations
-  manual: False
-  default: False
-
-flag static
-  description: Emit statically-linked binary
-  manual: False
-  default: False
-
 library
   exposed-modules:
       GaloisField
@@ -44,54 +34,65 @@
   hs-source-dirs:
       src
   default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses
-  ghc-options: -Wall
+  ghc-options: -O2 -Wall
   build-depends:
-      base >=4.7 && <5
+      MonadRandom
+    , base >=4.7 && <5
     , integer-gmp
     , protolude >=0.2
+    , tasty-quickcheck
+    , wl-pprint-text
   default-language: Haskell2010
 
-test-suite tests
+test-suite galois-field-tests
   type: exitcode-stdio-1.0
   main-is: Main.hs
   other-modules:
-      GaloisField
-      PrimeField
-      PolynomialRing
-      ExtensionField
+      ExtensionFieldTests
       GaloisFieldTests
-      PrimeFieldTests
       PolynomialRingTests
-      ExtensionFieldTests
+      PrimeFieldTests
+      ExtensionField
+      GaloisField
+      PolynomialRing
+      PrimeField
+      Paths_galois_field
   hs-source-dirs:
-      src
       tests
+      src
   default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses
+  ghc-options: -O2 -main-is Main
   build-depends:
-      base >=4.7 && <5
+      MonadRandom
+    , base >=4.7 && <5
     , integer-gmp
     , protolude >=0.2
     , tasty
     , tasty-discover
     , tasty-quickcheck
+    , wl-pprint-text
   default-language: Haskell2010
 
-benchmark benchmarks
+benchmark galois-field-benchmarks
   type: exitcode-stdio-1.0
   main-is: Main.hs
   other-modules:
+      ExtensionField
       GaloisField
-      PrimeField
       PolynomialRing
-      ExtensionField
-      Benchmarks
+      PrimeField
+      Paths_galois_field
   hs-source-dirs:
-      src
       benchmarks
+      src
   default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses
+  ghc-options: -O2 -main-is Main
   build-depends:
-      base >=4.7 && <5
+      MonadRandom
+    , base >=4.7 && <5
     , criterion
     , integer-gmp
     , protolude >=0.2
+    , tasty-quickcheck
+    , wl-pprint-text
   default-language: Haskell2010
diff --git a/src/ExtensionField.hs b/src/ExtensionField.hs
--- a/src/ExtensionField.hs
+++ b/src/ExtensionField.hs
@@ -9,53 +9,113 @@
 
 import Protolude
 
+import Control.Monad.Random (Random(..), getRandom)
+import Test.Tasty.QuickCheck (Arbitrary(..), choose, sized)
+import Text.PrettyPrint.Leijen.Text (Pretty(..))
+
 import GaloisField (GaloisField(..))
-import PolynomialRing (Polynomial(..), polyDiv, polyInv, toPoly)
+import PolynomialRing (Polynomial(..), cut, polyInv, polyMul, polyQR)
 
+-- | Irreducible monic splitting polynomial of extension field
+class IrreducibleMonic k im where
+  {-# MINIMAL split #-}
+  split :: ExtensionField k im -> Polynomial k -- ^ Splitting polynomial
+
 -- | Extension fields @GF(p^q)[X]/<f(X)>@ for @p@ prime, @q@ positive, and
 -- @f(X)@ irreducible monic in @GF(p^q)[X]@
 newtype ExtensionField k im = EF (Polynomial k)
   deriving (Eq, Generic, NFData, Show)
 
--- | Irreducible monic splitting polynomial of extension field
-class IrreducibleMonic k im where
-  split :: (k, im) -> Polynomial k
+-- | Extension fields are arbitrary
+instance (Arbitrary k, GaloisField k, IrreducibleMonic k im)
+  => Arbitrary (ExtensionField k im) where
+  arbitrary = fromList <$> sized (const poly)
+    where
+      poly = choose (1, length xs - 1) >>= mapM (const arbitrary) . enumFromTo 1
+        where
+          X xs = split (witness :: ExtensionField k im)
 
 -- | Extension fields are fields
 instance (GaloisField k, IrreducibleMonic k im)
   => Fractional (ExtensionField k im) where
-  recip (EF y)        = case polyInv y (split (witness :: (k, im))) of
-    Just z -> EF z
-    _      -> panic "no multiplicative inverse."
+  recip (EF (X ys))   = case polyInv ys xs of
+    Just zs -> EF (X zs)
+    _       -> panic "no multiplicative inverse."
+    where
+      X xs = split (witness :: ExtensionField k im)
   {-# INLINE recip #-}
   fromRational (y:%z) = fromInteger y / fromInteger z
+  {-# INLINABLE fromRational #-}
 
 -- | Extension fields are Galois fields
 instance (GaloisField k, IrreducibleMonic k im)
   => GaloisField (ExtensionField k im) where
-  char = const (char (witness :: k))
+  char          = const (char (witness :: k))
+  {-# INLINE char #-}
+  deg           = const (deg (witness :: k) * length xs - 1)
+    where
+      X xs = split (witness :: ExtensionField k im)
+  {-# INLINE deg #-}
+  pow y@(EF (X ys)) n
+    | n < 0     = pow (recip y) (-n)
+    | otherwise = EF (X (pow' [1] ys n))
+    where
+      X xs = split (witness :: ExtensionField k im)
+      mul = (.) (snd . flip polyQR xs) . polyMul
+      pow' ws zs m
+        | m == 0    = ws
+        | m == 1    = mul ws zs
+        | even m    = pow' ws (mul zs zs) (div m 2)
+        | otherwise = pow' (mul ws zs) (mul zs zs) (div m 2)
+  {-# INLINE pow #-}
+  rnd           = getRandom
+  {-# INLINE rnd #-}
 
 -- | Extension fields are rings
 instance (GaloisField k, IrreducibleMonic k im)
   => Num (ExtensionField k im) where
-  EF y + EF z   = EF (y + z)
-  EF y * EF z   = EF (snd (polyDiv (y * z) (split (witness :: (k, im)))))
-  EF y - EF z   = EF (y - z)
-  negate (EF y) = EF (-y)
-  fromInteger   = EF . fromInteger
-  abs           = panic "not implemented."
-  signum        = panic "not implemented."
+  EF y + EF z           = EF (y + z)
+  {-# INLINE (+) #-}
+  EF (X ys) * EF (X zs) = EF (X (snd (polyQR (polyMul ys zs) xs)))
+    where
+      X xs = split (witness :: ExtensionField k im)
+  {-# INLINE (*) #-}
+  EF y - EF z           = EF (y - z)
+  {-# INLINE (-) #-}
+  negate (EF y)         = EF (-y)
+  {-# INLINE negate #-}
+  fromInteger           = EF . fromInteger
+  {-# INLINABLE fromInteger #-}
+  abs                   = panic "not implemented."
+  signum                = panic "not implemented."
 
+-- | Extension fields are pretty
+instance (GaloisField k, IrreducibleMonic k im)
+  => Pretty (ExtensionField k im) where
+  pretty (EF y) = pretty y
+
+-- | Extension fields are random
+instance (GaloisField k, IrreducibleMonic k im)
+  => Random (ExtensionField k im) where
+  random  = first (EF . X . cut) . unfold (length xs - 1) []
+    where
+      X xs = split (witness :: ExtensionField k im)
+      unfold n ys g = if n <= 0 then (ys, g) else
+        let (y, g') = random g in unfold (n - 1) (y : ys) g'
+  randomR = panic "not implemented."
+
 -- | List from field
 fromField :: ExtensionField k im -> [k]
-fromField (EF (X ks)) = ks
-{-# INLINE fromField #-}
+fromField (EF (X xs)) = xs
+{-# INLINABLE fromField #-}
 
 -- | Field from list
 fromList :: forall k im . (GaloisField k, IrreducibleMonic k im)
   => [k] -> ExtensionField k im
-fromList = EF . snd . flip polyDiv (split (witness :: (k, im))) . toPoly
-{-# INLINE fromList #-}
+fromList = EF . X . snd . flip polyQR xs . cut
+  where
+    X xs = split (witness :: ExtensionField k im)
+{-# INLINABLE fromList #-}
 
 -- | Current indeterminate variable
 x :: GaloisField k => Polynomial k
diff --git a/src/GaloisField.hs b/src/GaloisField.hs
--- a/src/GaloisField.hs
+++ b/src/GaloisField.hs
@@ -4,8 +4,26 @@
 
 import Protolude
 
+import Control.Monad.Random (MonadRandom, Random)
+import Test.Tasty.QuickCheck (Arbitrary)
+import Text.PrettyPrint.Leijen.Text (Pretty)
+
 -- | Galois fields @GF(p^q)@ for @p@ prime and @q@ non-negative
-class (Eq k, Fractional k, Show k) => GaloisField k where
-  {-# MINIMAL char #-}
-  -- | Characteristic of field
-  char :: k -> Integer
+class (Arbitrary k, Eq k, Fractional k, Pretty k, Random k, Show k)
+  => GaloisField k where
+  {-# MINIMAL char, deg, pow, rnd #-}
+
+  -- Characteristics
+  char :: k -> Integer  -- ^ Characteristic @q@ of field
+
+  deg :: k -> Int       -- ^ Degree @q@ of field
+
+  order :: k -> Integer -- ^ Order @p^q@ of field
+  order = (^) <$> char <*> deg
+  {-# INLINE order #-}
+
+  -- Functions
+  pow :: k -> Integer -> k -- @x@ to the power of @y@
+
+  -- Randomisation
+  rnd :: MonadRandom m => m k -- ^ Random element of field
diff --git a/src/PolynomialRing.hs b/src/PolynomialRing.hs
--- a/src/PolynomialRing.hs
+++ b/src/PolynomialRing.hs
@@ -1,97 +1,104 @@
 module PolynomialRing
   ( Polynomial(..)
-  , degree
-  , last
-  , polyDiv
+  , cut
   , polyInv
+  , polyMul
+  , polyQR
   , toPoly
   ) where
 
 import Protolude
 
+import Test.Tasty.QuickCheck (Arbitrary(..))
+import Text.PrettyPrint.Leijen.Text (Pretty(..))
+
 import GaloisField (GaloisField(..))
 
--- | Polynomial rings
 newtype Polynomial k = X [k]
   deriving (Eq, Generic, NFData, Show)
 
--- | Polynomial rings are rings
+instance (Arbitrary k, GaloisField k) => Arbitrary (Polynomial k) where
+  arbitrary = toPoly <$> arbitrary
+
 instance GaloisField k => Num (Polynomial k) where
   X xs + X ys   = X (polyAdd xs ys)
+  {-# INLINE (+) #-}
   X xs * X ys   = X (polyMul xs ys)
+  {-# INLINE (*) #-}
+  X xs - X ys   = X (polySub xs ys)
+  {-# INLINE (-) #-}
   negate (X xs) = X (map negate xs)
-  fromInteger   = fromInt
+  {-# INLINE negate #-}
+  fromInteger n = X (let m = fromInteger n in if m == 0 then [] else [m])
+  {-# INLINABLE fromInteger #-}
   abs           = panic "not implemented."
   signum        = panic "not implemented."
 
--- | Polynomial addition
+instance GaloisField k => Pretty (Polynomial k) where
+  pretty (X xs) = pretty xs
+
 polyAdd :: GaloisField k => [k] -> [k] -> [k]
-polyAdd []     xs     = xs
 polyAdd xs     []     = xs
-polyAdd (x:xs) (y:ys) = if z == 0 && null zs then [] else z : zs
-  where
-    z  = x + y
-    zs = polyAdd xs ys
+polyAdd []     ys     = ys
+polyAdd (x:xs) (y:ys) = let z  = x + y
+                            zs = polyAdd xs ys
+                        in if z == 0 && null zs then [] else z : zs
 {-# INLINE polyAdd #-}
 
--- | Polynomial multiplication
 polyMul :: GaloisField k => [k] -> [k] -> [k]
-polyMul []     _  = []
 polyMul _      [] = []
-polyMul (x:xs) ys = if null xs then ws else polyAdd ws (0 : zs)
-  where
-    ws = map (* x) ys
-    zs = polyMul xs ys
+polyMul []     _  = []
+polyMul (x:xs) ys = let ws = map (* x) ys
+                        zs = polyMul xs ys
+                    in if null xs then ws else polyAdd ws (0 : zs)
 {-# INLINE polyMul #-}
 
--- | Polynomial from integer
-fromInt :: GaloisField k => Integer -> Polynomial k
-fromInt n = X (let m = fromInteger n in if m == 0 then [] else [m])
-{-# INLINE fromInt #-}
-
--- | Polynomial degree
-degree :: Polynomial k -> Int
-degree (X xs) = length xs
-{-# INLINE degree #-}
-
--- | Polynomial leading coefficient
-last :: GaloisField k => Polynomial k -> k
-last (X [])     = 0
-last (X [x])    = x
-last (X (_:xs)) = last (X xs)
-{-# INLINE last #-}
+polySub :: GaloisField k => [k] -> [k] -> [k]
+polySub xs     []     = xs
+polySub []     ys     = map negate ys
+polySub (x:xs) (y:ys) = let z  = x - y
+                            zs = polySub xs ys
+                        in if z == 0 && null zs then [] else z : zs
+{-# INLINE polySub #-}
 
--- | Polynomial division
-polyDiv :: forall k . GaloisField k
-  => Polynomial k -> Polynomial k -> (Polynomial k, Polynomial k)
-polyDiv ns ds = polyQR (0, ns)
+polyQR :: forall k . GaloisField k => [k] -> [k] -> ([k], [k])
+polyQR xs ys = polyGCD ([], xs)
   where
-    polyQR :: (Polynomial k, Polynomial k) -> (Polynomial k, Polynomial k)
-    polyQR qr@(qs, rs)
-      | degree rs < degree ds = qr
-      | otherwise             = polyQR (qs + ts, rs - ts * ds)
+    m :: Int
+    m = length ys
+    polyGCD :: ([k], [k]) -> ([k], [k])
+    polyGCD qr@(qs, rs)
+      | n < 0     = qr
+      | otherwise = polyGCD (polyAdd qs ts, polySub rs (polyMul ts ys))
       where
-        ts = X (replicate (degree rs - degree ds) 0 ++ [last rs / last ds])
-{-# INLINE polyDiv #-}
+        n :: Int
+        n = length rs - m
+        ts :: [k]
+        ts = replicate n 0 ++ [last rs / last ys]
+        last :: [k] -> k
+        last []     = 0
+        last [z]    = z
+        last (_:zs) = last zs
+{-# INLINE polyQR #-}
 
--- | Polynomial inversion
-polyInv :: forall k . GaloisField k
-  => Polynomial k -> Polynomial k -> Maybe (Polynomial k)
-polyInv (X [x]) _ = Just (X [recip x])
-polyInv xs ps     = case extGCD ps xs of
-  (X [y], (X ys, _)) -> Just (X (map (/ y) ys))
-  _                  -> Nothing
+polyInv :: forall k . GaloisField k => [k] -> [k] -> Maybe [k]
+polyInv [x] _ = Just [recip x]
+polyInv xs ps = case extGCD (ps, xs) of
+  ([y], (ys, _)) -> Just (map (/ y) ys)
+  _              -> Nothing
   where
-    extGCD :: Polynomial k -> Polynomial k
-      -> (Polynomial k, (Polynomial k, Polynomial k))
-    extGCD y 0 = (y, (0, 1))
-    extGCD y x = (g, (t - s * q, s))
+    extGCD :: ([k], [k]) -> ([k], ([k], [k]))
+    extGCD (y, []) = (y, ([], [1]))
+    extGCD (y, x)  = (g, (polySub t (polyMul s q), s))
       where
-        (q, r)      = polyDiv y x
-        (g, (s, t)) = extGCD x r
+        (q, r)      = polyQR y x
+        (g, (s, t)) = extGCD (x, r)
 {-# INLINE polyInv #-}
 
--- | List to polynomial
+cut :: GaloisField k => [k] -> [k]
+cut = reverse . dropWhile (== 0) . reverse
+{-# INLINABLE cut #-}
+
 toPoly :: GaloisField k => [k] -> Polynomial k
-toPoly = X . reverse . dropWhile (== 0) . reverse
-{-# INLINE toPoly #-}
+toPoly = X . cut
+{-# INLINABLE toPoly #-}
diff --git a/src/PrimeField.hs b/src/PrimeField.hs
--- a/src/PrimeField.hs
+++ b/src/PrimeField.hs
@@ -5,45 +5,72 @@
 
 import Protolude
 
-import GHC.Integer.GMP.Internals (recipModInteger)
+import Control.Monad.Random (Random(..), getRandom)
+import GHC.Integer.GMP.Internals (powModInteger, recipModInteger)
+import Test.Tasty.QuickCheck (Arbitrary(..))
+import Text.PrettyPrint.Leijen.Text (Pretty(..))
 
 import GaloisField (GaloisField(..))
 
 -- | Prime fields @GF(p)@ for @p@ prime
 newtype PrimeField (p :: Nat) = PF Integer
-  deriving (Eq, Generic, NFData, Show)
+  deriving (Bits, Eq, Generic, NFData, Show)
 
+-- | Prime fields are arbitrary
+instance KnownNat p => Arbitrary (PrimeField p) where
+  arbitrary = fromInteger <$> arbitrary
+
 -- | Prime fields are fields
 instance KnownNat p => Fractional (PrimeField p) where
   recip y@(PF x)      = PF (recipModInteger x (natVal y))
+  {-# INLINE recip #-}
   fromRational (x:%y) = fromInteger x / fromInteger y
+  {-# INLINABLE fromRational #-}
 
 -- | Prime fields are Galois fields
 instance KnownNat p => GaloisField (PrimeField p) where
-  char = natVal
+  char           = natVal
+  {-# INLINE char #-}
+  deg            = const 1
+  {-# INLINE deg #-}
+  pow y@(PF x) n = PF (powModInteger x n (natVal y))
+  {-# INLINE pow #-}
+  rnd            = getRandom
+  {-# INLINE rnd #-}
 
 -- | Prime fields are rings
 instance KnownNat p => Num (PrimeField p) where
-  z@(PF x) + PF y = PF (if xy >= p then xy - p else xy)
+  z@(PF x) + PF y = PF (if xyp >= 0 then xyp else xy)
     where
-      xy = x + y
-      p  = natVal z
+      xy  = x + y
+      xyp = xy - natVal z
   {-# INLINE (+) #-}
   z@(PF x) * PF y = PF (rem (x * y) (natVal z))
+  {-# INLINE (*) #-}
   z@(PF x) - PF y = PF (if xy >= 0 then xy else xy + natVal z)
     where
       xy = x - y
   {-# INLINE (-) #-}
   negate y@(PF x) = PF (if x == 0 then 0 else -x + natVal y)
+  {-# INLINE negate #-}
   fromInteger x   = PF (if y >= 0 then y else y + p)
     where
       y = rem x p
       p = natVal (witness :: PrimeField p)
-  {-# INLINE fromInteger #-}
+  {-# INLINABLE fromInteger #-}
   abs             = panic "not implemented."
   signum          = panic "not implemented."
 
+-- | Prime fields are pretty
+instance KnownNat p => Pretty (PrimeField p) where
+  pretty (PF x) = pretty [x]
+
+-- | Prime fields are random
+instance KnownNat p => Random (PrimeField p) where
+  random  = first PF . randomR (0, natVal (witness :: PrimeField p) - 1)
+  randomR = panic "not implemented."
+
 -- | Embed to integers
 toInt :: PrimeField p -> Integer
 toInt (PF x) = x
-{-# INLINE toInt #-}
+{-# INLINABLE toInt #-}
diff --git a/tests/ExtensionFieldTests.hs b/tests/ExtensionFieldTests.hs
--- a/tests/ExtensionFieldTests.hs
+++ b/tests/ExtensionFieldTests.hs
@@ -2,82 +2,119 @@
 
 import Protolude
 
+import ExtensionField
 import Test.Tasty
-import Test.Tasty.QuickCheck
 
-import ExtensionField
-import GaloisField
 import GaloisFieldTests
-import PolynomialRing
-import PolynomialRingTests
-import PrimeField
 import PrimeFieldTests
 
-instance (Arbitrary k, GaloisField k, IrreducibleMonic k ps)
-  => Arbitrary (ExtensionField k ps) where
-  arbitrary = fromList <$> sized (const poly)
-    where
-      poly = choose (1, degree (split (witness :: (k, ps))) - 1)
-        >>= mapM (const arbitrary) . enumFromTo 1
+data P11
+instance IrreducibleMonic FS2 P11 where
+  split _ = x^2 + x + 1
+type FS4 = ExtensionField FS2 P11
+test_S4 :: TestTree
+test_S4 = fieldAxioms (Proxy :: Proxy FS4) "FS4"
 
-type F2 = PrimeField 2
-data P11; instance IrreducibleMonic F2 P11 where split _ = x^2 + x + 1
-test_4 = fieldAxioms (Proxy :: Proxy (ExtensionField F2 P11)) "F4"
-data P110; instance IrreducibleMonic F2 P110 where split _ = x^3 + x + 1
-test_8 = fieldAxioms (Proxy :: Proxy (ExtensionField F2 P110)) "F8"
-data P101; instance IrreducibleMonic F2 P101 where split _ = x^3 + x^2 + 1
-test_8' = fieldAxioms (Proxy :: Proxy (ExtensionField F2 P110)) "F8'"
+data P110
+instance IrreducibleMonic FS2 P110 where
+  split _ = x^3 + x + 1
+type FS8 = ExtensionField FS2 P110
+test_S8 :: TestTree
+test_S8 = fieldAxioms (Proxy :: Proxy FS8) "FS8"
 
-type F3 = PrimeField 3
-data P10; instance IrreducibleMonic F3 P10 where split _ = x^2 + 1
-test_9 = fieldAxioms (Proxy :: Proxy (ExtensionField F3 P10)) "F9"
-data P21; instance IrreducibleMonic F3 P21 where split _ = x^2 + x - 1
-test_9' = fieldAxioms (Proxy :: Proxy (ExtensionField F3 P21)) "F9'"
-data P22; instance IrreducibleMonic F3 P22 where split _ = x^2 - x - 1
-test_9'' = fieldAxioms (Proxy :: Proxy (ExtensionField F3 P22)) "F9''"
+data P101
+instance IrreducibleMonic FS2 P101 where
+  split _ = x^3 + x^2 + 1
+type FS8' = ExtensionField FS2 P101
+test_S8' :: TestTree
+test_S8' = fieldAxioms (Proxy :: Proxy FS8') "FS8'"
 
-type FA = PrimeField 2147483647
-instance IrreducibleMonic FA P10 where split _ = x^2 + 1
-test_A = fieldAxioms (Proxy :: Proxy (ExtensionField FA P10)) "FA2"
+data P10
+instance IrreducibleMonic FS3 P10 where
+  split _ = x^2 + 1
+type FS9 = ExtensionField FS3 P10
+test_S9 :: TestTree
+test_S9 = fieldAxioms (Proxy :: Proxy FS9) "FS9"
 
-type FB = PrimeField 2305843009213693951
-instance IrreducibleMonic FB P10 where split _ = x^2 + 1
-test_B = fieldAxioms (Proxy :: Proxy (ExtensionField FB P10)) "FB2"
+data P21
+instance IrreducibleMonic FS3 P21 where
+  split _ = x^2 + x - 1
+type FS9' = ExtensionField FS3 P21
+test_S9' :: TestTree
+test_S9' = fieldAxioms (Proxy :: Proxy FS9') "FS9'"
 
-type FC = PrimeField 618970019642690137449562111
-instance IrreducibleMonic FC P10 where split _ = x^2 + 1
-test_C = fieldAxioms (Proxy :: Proxy (ExtensionField FC P10)) "FC2"
+data P22
+instance IrreducibleMonic FS3 P22 where
+  split _ = x^2 - x - 1
+type FS9'' = ExtensionField FS3 P22
+test_S9'' :: TestTree
+test_S9'' = fieldAxioms (Proxy :: Proxy FS9'') "FS9''"
 
-type FD = PrimeField 162259276829213363391578010288127
-instance IrreducibleMonic FD P10 where split _ = x^2 + 1
-test_D = fieldAxioms (Proxy :: Proxy (ExtensionField FD P10)) "FD2"
+instance IrreducibleMonic FM0 P10 where
+  split _ = x^2 + 1
+type FL0 = ExtensionField FM0 P10
+test_L0 :: TestTree
+test_L0 = fieldAxioms (Proxy :: Proxy FL0) "FL0"
 
-type FE = PrimeField 170141183460469231731687303715884105727
-instance IrreducibleMonic FE P10 where split _ = x^2 + 1
-test_E = fieldAxioms (Proxy :: Proxy (ExtensionField FE P10)) "FE2"
+instance IrreducibleMonic FM1 P10 where
+  split _ = x^2 + 1
+type FL1 = ExtensionField FM1 P10
+test_L1 :: TestTree
+test_L1 = fieldAxioms (Proxy :: Proxy FL1) "FL1"
 
-type FV = PrimeField 20988936657440586486151264256610222593863921
-instance IrreducibleMonic FV P10 where split _ = x^2 + 1
-test_V = fieldAxioms (Proxy :: Proxy (ExtensionField FV P10)) "FV2"
+instance IrreducibleMonic FM2 P10 where
+  split _ = x^2 + 1
+type FL2 = ExtensionField FM2 P10
+test_L2 :: TestTree
+test_L2 = fieldAxioms (Proxy :: Proxy FL2) "FL2"
 
-type FX = PrimeField 5210644015679228794060694325390955853335898483908056458352183851018372555735221
-instance IrreducibleMonic FX P10 where split _ = x^2 + 1
-test_X = fieldAxioms (Proxy :: Proxy (ExtensionField FX P10)) "FX2"
+instance IrreducibleMonic FM3 P10 where
+  split _ = x^2 + 1
+type FL3 = ExtensionField FM3 P10
+test_L3 :: TestTree
+test_L3 = fieldAxioms (Proxy :: Proxy FL3) "FL3"
 
-type FZ = PrimeField 741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737
-instance IrreducibleMonic FZ P10 where split _ = x^2 + 1
-test_Z = fieldAxioms (Proxy :: Proxy (ExtensionField FZ P10)) "FZ2"
+instance IrreducibleMonic FM4 P10 where
+  split _ = x^2 + 1
+type FL4 = ExtensionField FM4 P10
+test_L4 :: TestTree
+test_L4 = fieldAxioms (Proxy :: Proxy FL4) "FL4"
 
-type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583
+instance IrreducibleMonic FVL P10 where
+  split _ = x^2 + 1
+type FV2 = ExtensionField FVL P10
+test_V2 :: TestTree
+test_V2 = fieldAxioms (Proxy :: Proxy FV2) "FV2"
+
+instance IrreducibleMonic FXL P10 where
+  split _ = x^2 + 1
+type FX2 = ExtensionField FXL P10
+test_X2 :: TestTree
+test_X2 = fieldAxioms (Proxy :: Proxy FX2) "FX2"
+
+instance IrreducibleMonic FZL P10 where
+  split _ = x^2 + 1
+type FZ2 = ExtensionField FZL P10
+test_Z2 :: TestTree
+test_Z2 = fieldAxioms (Proxy :: Proxy FZ2) "FZ2"
+
 data Pu
-instance IrreducibleMonic Fq Pu where split _ = x^2 + 1
+instance IrreducibleMonic Fq Pu where
+  split _ = x^2 + 1
 type Fq2 = ExtensionField Fq Pu
+test_Fq2 :: TestTree
 test_Fq2 = fieldAxioms (Proxy :: Proxy Fq2) "Fq2"
+
 data Pv
-instance IrreducibleMonic Fq2 Pv where split _ = x^3 - (9 + t x)
+instance IrreducibleMonic Fq2 Pv where
+  split _ = x^3 - (9 + t x)
 type Fq6 = ExtensionField Fq2 Pv
+test_Fq6 :: TestTree
 test_Fq6 = fieldAxioms (Proxy :: Proxy Fq6) "Fq6"
+
 data Pw
-instance IrreducibleMonic Fq6 Pw where split _ = x^2 - t x
+instance IrreducibleMonic Fq6 Pw where
+  split _ = x^2 - t x
 type Fq12 = ExtensionField Fq6 Pw
+test_Fq12 :: TestTree
 test_Fq12 = fieldAxioms (Proxy :: Proxy Fq12) "Fq12"
diff --git a/tests/PolynomialRingTests.hs b/tests/PolynomialRingTests.hs
--- a/tests/PolynomialRingTests.hs
+++ b/tests/PolynomialRingTests.hs
@@ -2,37 +2,44 @@
 
 import Protolude
 
-import Test.Tasty.QuickCheck
-
-import GaloisField
-import GaloisFieldTests
 import PolynomialRing
-import PrimeField
-import PrimeFieldTests
+import Test.Tasty
 
-instance (Arbitrary k, GaloisField k) => Arbitrary (Polynomial k) where
-  arbitrary = toPoly <$> arbitrary
+import PrimeFieldTests
+import GaloisFieldTests
 
-test_S2X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 2))) "FS2[X]"
+test_S2X :: TestTree
+test_S2X = ringAxioms (Proxy :: Proxy (Polynomial FS2)) "FS2[X]"
 
-test_S3X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 3))) "FS3[X]"
+test_S3X :: TestTree
+test_S3X = ringAxioms (Proxy :: Proxy (Polynomial FS3)) "FS3[X]"
 
-test_S5X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 5))) "FS5[X]"
+test_S5X :: TestTree
+test_S5X = ringAxioms (Proxy :: Proxy (Polynomial FS5)) "FS5[X]"
 
-test_S7X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 7))) "FS7[X]"
+test_S7X :: TestTree
+test_S7X = ringAxioms (Proxy :: Proxy (Polynomial FS7)) "FS7[X]"
 
-test_M0X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 2147483647))) "FM0[X]"
+test_M0X :: TestTree
+test_M0X = ringAxioms (Proxy :: Proxy (Polynomial FM0)) "FM0[X]"
 
-test_M1X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 2305843009213693951))) "FM1[X]"
+test_M1X :: TestTree
+test_M1X = ringAxioms (Proxy :: Proxy (Polynomial FM1)) "FM1[X]"
 
-test_M2X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 618970019642690137449562111))) "FM2[X]"
+test_M2X :: TestTree
+test_M2X = ringAxioms (Proxy :: Proxy (Polynomial FM2)) "FM2[X]"
 
-test_M3X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 162259276829213363391578010288127))) "FM3[X]"
+test_M3X :: TestTree
+test_M3X = ringAxioms (Proxy :: Proxy (Polynomial FM3)) "FM3[X]"
 
-test_M4X = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 170141183460469231731687303715884105727))) "FM4[X]"
+test_M4X :: TestTree
+test_M4X = ringAxioms (Proxy :: Proxy (Polynomial FM4)) "FM4[X]"
 
-test_VLX = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 20988936657440586486151264256610222593863921))) "FVL[X]"
+test_VLX :: TestTree
+test_VLX = ringAxioms (Proxy :: Proxy (Polynomial FVL)) "FVL[X]"
 
-test_XLX = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 5210644015679228794060694325390955853335898483908056458352183851018372555735221))) "FXL[X]"
+test_XLX :: TestTree
+test_XLX = ringAxioms (Proxy :: Proxy (Polynomial FXL)) "FXL[X]"
 
-test_ZLX = ringAxioms (Proxy :: Proxy (Polynomial (PrimeField 741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737))) "FZL[X]"
+test_ZLX :: TestTree
+test_ZLX = ringAxioms (Proxy :: Proxy (Polynomial FZL)) "FZL[X]"
diff --git a/tests/PrimeFieldTests.hs b/tests/PrimeFieldTests.hs
--- a/tests/PrimeFieldTests.hs
+++ b/tests/PrimeFieldTests.hs
@@ -2,34 +2,61 @@
 
 import Protolude
 
-import Test.Tasty.QuickCheck
-
 import PrimeField
+import Test.Tasty
+
 import GaloisFieldTests
 
-instance KnownNat p => Arbitrary (PrimeField p) where
-  arbitrary = fromInteger <$> arbitrary
+type FS2 = PrimeField 2
+type FS3 = PrimeField 3
+type FS5 = PrimeField 5
+type FS7 = PrimeField 7
 
-test_S2 = fieldAxioms (Proxy :: Proxy (PrimeField 2)) "FS2"
+type FM0 = PrimeField 2147483647
+type FM1 = PrimeField 2305843009213693951
+type FM2 = PrimeField 618970019642690137449562111
+type FM3 = PrimeField 162259276829213363391578010288127
+type FM4 = PrimeField 170141183460469231731687303715884105727
 
-test_S3 = fieldAxioms (Proxy :: Proxy (PrimeField 3)) "FS3"
+type FVL = PrimeField 20988936657440586486151264256610222593863921
+type FXL = PrimeField 5210644015679228794060694325390955853335898483908056458352183851018372555735221
+type FZL = PrimeField 741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737
 
-test_S5 = fieldAxioms (Proxy :: Proxy (PrimeField 5)) "FS5"
+type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583
 
-test_S7 = fieldAxioms (Proxy :: Proxy (PrimeField 7)) "FS7"
 
-test_M0 = fieldAxioms (Proxy :: Proxy (PrimeField 2147483647)) "FM0"
+test_S2 :: TestTree
+test_S2 = fieldAxioms (Proxy :: Proxy FS2) "FS2"
 
-test_M1 = fieldAxioms (Proxy :: Proxy (PrimeField 2305843009213693951)) "FM1"
+test_S3 :: TestTree
+test_S3 = fieldAxioms (Proxy :: Proxy FS3) "FS3"
 
-test_M2 = fieldAxioms (Proxy :: Proxy (PrimeField 618970019642690137449562111)) "FM2"
+test_S5 :: TestTree
+test_S5 = fieldAxioms (Proxy :: Proxy FS5) "FS5"
 
-test_M3 = fieldAxioms (Proxy :: Proxy (PrimeField 162259276829213363391578010288127)) "FM3"
+test_S7 :: TestTree
+test_S7 = fieldAxioms (Proxy :: Proxy FS7) "FS7"
 
-test_M4 = fieldAxioms (Proxy :: Proxy (PrimeField 170141183460469231731687303715884105727)) "FM4"
+test_M0 :: TestTree
+test_M0 = fieldAxioms (Proxy :: Proxy FM0) "FM0"
 
-test_VL = fieldAxioms (Proxy :: Proxy (PrimeField 20988936657440586486151264256610222593863921)) "FVL"
+test_M1 :: TestTree
+test_M1 = fieldAxioms (Proxy :: Proxy FM1) "FM1"
 
-test_XL = fieldAxioms (Proxy :: Proxy (PrimeField 5210644015679228794060694325390955853335898483908056458352183851018372555735221)) "FXL"
+test_M2 :: TestTree
+test_M2 = fieldAxioms (Proxy :: Proxy FM2) "FM2"
 
-test_ZL = fieldAxioms (Proxy :: Proxy (PrimeField 741640062627530801524787141901937474059940781097519023905821316144415759504705008092818711693940737)) "FZL"
+test_M3 :: TestTree
+test_M3 = fieldAxioms (Proxy :: Proxy FM3) "FM3"
+
+test_M4 :: TestTree
+test_M4 = fieldAxioms (Proxy :: Proxy FM4) "FM4"
+
+test_VL :: TestTree
+test_VL = fieldAxioms (Proxy :: Proxy FVL) "FVL"
+
+test_XL :: TestTree
+test_XL = fieldAxioms (Proxy :: Proxy FXL) "FXL"
+
+test_ZL :: TestTree
+test_ZL = fieldAxioms (Proxy :: Proxy FZL) "FZL"