HaskellForMaths-0.3.4: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Core.Field

Description

A module defining the field Q of rationals and the small finite fields F2, F3, F4, F5, F7, F8, F9, F11, F13, F16, F17, F19, F23, F25.

Given a prime power q, Fq is the type representing elements of the field (eg `F4`), fq is a list of the elements of the field, beginning 0,1,... (eg `f4`), and for prime power fields, aq is a primitive element, which generates the multiplicative group (eg `a4`).

The design philosophy is that fq, the list of elements, represents the field. Thus, many functions elsewhere in the library expect to take fq as an argument, telling them which field to work over.

Synopsis

# Documentation

newtype Q Source

Q is just the rationals, but with a better show function than the Prelude version

Constructors

 Q Rational

Instances

 Eq Q Fractional Q Num Q Ord Q Show Q

newtype F2 Source

F2 is a type for the finite field with 2 elements

Constructors

 F2 Int

Instances

 Eq F2 Fractional F2 Num F2 Ord F2 Show F2

f2 :: [F2]Source

f2 is a list of the elements of F2

newtype F3 Source

F3 is a type for the finite field with 3 elements

Constructors

 F3 Int

Instances

 Eq F3 Fractional F3 Num F3 Ord F3 Show F3

f3 :: [F3]Source

f3 is a list of the elements of F3

newtype F5 Source

F5 is a type for the finite field with 5 elements

Constructors

 F5 Int

Instances

 Eq F5 Fractional F5 Num F5 Ord F5 Show F5

f5 :: [F5]Source

f5 is a list of the elements of F5

newtype F7 Source

F7 is a type for the finite field with 7 elements

Constructors

 F7 Int

Instances

 Eq F7 Fractional F7 Num F7 Ord F7 Show F7

f7 :: [F7]Source

f7 is a list of the elements of F7

newtype F11 Source

F11 is a type for the finite field with 11 elements

Constructors

 F11 Int

Instances

 Eq F11 Fractional F11 Num F11 Ord F11 Show F11

f11 :: [F11]Source

f11 is a list of the elements of F11

newtype F13 Source

F13 is a type for the finite field with 13 elements

Constructors

 F13 Int

Instances

 Eq F13 Fractional F13 Num F13 Ord F13 Show F13

f13 :: [F13]Source

f13 is a list of the elements of F13

newtype F17 Source

F17 is a type for the finite field with 17 elements

Constructors

 F17 Int

Instances

 Eq F17 Fractional F17 Num F17 Ord F17 Show F17

f17 :: [F17]Source

f17 is a list of the elements of F17

newtype F19 Source

F19 is a type for the finite field with 19 elements

Constructors

 F19 Int

Instances

 Eq F19 Fractional F19 Num F19 Ord F19 Show F19

f19 :: [F19]Source

f19 is a list of the elements of F19

newtype F23 Source

F23 is a type for the finite field with 23 elements

Constructors

 F23 Int

Instances

 Eq F23 Fractional F23 Num F23 Ord F23 Show F23

f23 :: [F23]Source

f23 is a list of the elements of F23

newtype F4 Source

F4 is a type for the finite field with 4 elements. F4 is represented as the extension of F2 by an element a4 satisfying x^2+x+1 = 0

Constructors

 F4 Int

Instances

 Eq F4 Fractional F4 Num F4 Ord F4 Show F4

a4 is a primitive element for F4 as an extension over F2. a4 satisfies x^2+x+1 = 0.

f4 :: [F4]Source

f4 is a list of the elements of F4

newtype F8 Source

F8 is a type for the finite field with 8 elements. F8 is represented as the extension of F2 by an element a8 satisfying x^3+x+1 = 0

Constructors

 F8 Int

Instances

 Eq F8 Fractional F8 Num F8 Ord F8 Show F8

a8 is a primitive element for F8 as an extension over F2. a8 satisfies x^3+x+1 = 0.

f8 :: [F8]Source

f8 is a list of the elements of F8

newtype F9 Source

F9 is a type for the finite field with 9 elements. F9 is represented as the extension of F3 by an element a9 satisfying x^2+2x+2 = 0

Constructors

 F9 Int

Instances

 Eq F9 Fractional F9 Num F9 Ord F9 Show F9

a9 is a primitive element for F9 as an extension over F3. a9 satisfies x^2+2x+2 = 0.

f9 :: [F9]Source

f9 is a list of the elements of F9

newtype F16 Source

F16 is a type for the finite field with 16 elements. F16 is represented as the extension of F2 by an element a16 satisfying x^4+x+1 = 0

Constructors

 F16 Int

Instances

 Eq F16 Fractional F16 Num F16 Ord F16 Show F16

a16 is a primitive element for F16 as an extension over F2. a16 satisfies x^4+x+1 = 0.

f16 :: [F16]Source

f16 is a list of the elements of F16

newtype F25 Source

F25 is a type for the finite field with 25 elements. F25 is represented as the extension of F5 by an element a25 satisfying x^2+4x+2 = 0

Constructors

 F25 Int

Instances

 Eq F25 Fractional F25 Num F25 Ord F25 Show F25

a25 is a primitive element for F25 as an extension over F5. a25 satisfies x^2+4x+2 = 0.

f25 :: [F25]Source

f25 is a list of the elements of F25