# OpenMath Content Dictionary: field3

Canonical URL:
http://www.openmath.org/cd/field3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
field3.ocd
CD as XML Encoded OpenMath:
field3.omcd
Defines:
field_by_poly, fraction_field, free_field
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

A CD of functions for basic constructions in field theory.

Written by Arjeh M. Cohen 2004-02-25


## free_field

Description:

This symbol represents a binary function. The first argument should be a natural number p which is zero or a prime number, the second argument a list or a set L. When evaluated on such arguments p and L, the function represents the field of rational functions in L over the rationals if p = 0 and over the field of integers mod p if p is a prime.

Example:
The rational function field Q(a,b) in the indeterminates a, b is
$\mathrm{free_field}\left(0,\left(a,b\right)\right)$
Signatures:
sts

 [Next: fraction_field] [Last: field_by_poly] [Top]

## fraction_field

Description:

This is a unary function. Its argument should be a domain (as in CD ring4). It denotes the fraction field of the domain.

Example:
The rationals equals fraction_field(Integers)
$Q=\mathrm{fraction_field}\left(Z\right)$
Signatures:
sts

 [Next: field_by_poly] [Previous: free_field] [Top]

## field_by_poly

Description:

This symbol is a binary function whose first argument is a univariate polynomial ring R over a field, and whose second argument is an irreducible polynomial f in this polynomial ring R. So, when applied to R and f, the function has value the quotient ring R/(f).

Example:
The finite field GF(2)[X]/(X^2+X+1) is represented by
$\left(\mathrm{field_by_poly}\left(\mathrm{poly_ring_d_named}\left({\mathbb{GF}}_{2},X\right),\mathrm{DMP}\left(\mathrm{poly_ring_d_named}\left({\mathbb{GF}}_{2},X\right),\mathrm{SDMP}\left(\mathrm{term}\left(1,0\right),\mathrm{term}\left(1,1\right),\mathrm{term}\left(1,2\right)\right)\right)\right)\right)\left(\right)$
or by
$\mathrm{field_by_poly}\left(\mathrm{poly_ring_d_named}\left({\mathbb{GF}}_{2},X\right),\mathrm{expression}\left(\mathrm{poly_ring_d_named}\left({\mathbb{GF}}_{2},X\right),1+X+{X}^{2}\right)\right)$
Signatures:
sts

 [First: free_field] [Previous: fraction_field] [Top]