# OpenMath Content Dictionary: field4

Canonical URL:
http://www.openmath.org/cd/field4.ocd
CD Base:
http://www.openmath.org/cd
CD File:
field4.ocd
CD as XML Encoded OpenMath:
field4.omcd
Defines:
automorphism_group, field_by_poly_map, field_by_poly_vector, homomorphism_by_generators
Date:
2004-06-01
Version:
1 (Revision 1)
Review Date:
2006-06-01
Status:
experimental

A CD of functions for morphisms of fields.

Written by Arjeh M. Cohen 2004-07-07


## automorphism_group

Description:

This is a function with a single argument which must be a field. It refers to the automorphism group of its argument.

Signatures:
sts

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## homomorphism_by_generators

Description:

This is a function with three arguments the first two of which must be fields F and K. The third argument should be a set or a list L of ordered pairs (lists of length 2). Each pair [x,y] from L consists of an element x from F and an element y from K. when applied to F, K, and L, the symbol represents the homomorphism from F to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.

Signatures:
sts

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## field_by_poly_map

Description:

Same as quotient_by_poly_map in CD ring5, except that R and the quotient ring R[X]/(f) are fields (so f is irreducible in R[X]).

Example:
An element aX + b of the finite field GF(3)[X]/(X^2+1) is represented by
$\left(\mathrm{field_by_poly_map}\left(\mathrm{poly_ring_d}\left({\mathbb{GF}}_{3},1\right),\mathrm{DMP}\left(\mathrm{poly_ring_d}\left({\mathbb{GF}}_{3},1\right),\mathrm{SDMP}\left(\mathrm{term}\left(1,0\right),\mathrm{term}\left(1,2\right)\right)\right)\right)\right)\left(\mathrm{DMP}\left(\mathrm{poly_ring_d}\left({\mathbb{GF}}_{3},1\right),\mathrm{SDMP}\left(\mathrm{term}\left(b,0\right)\right),\mathrm{term}\left(a,1\right)\right)\right)$
Signatures:
sts

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## field_by_poly_vector

Description:

This symbol is a binary function. Its first argument should be a field_by_poly(R,f). Its second argument should be a list L of elements of F, the coefficient field of the univariate polynomial ring R = F[X]. The length of the list L should be equal to the degree d of f. When applied to R and L, it represents the element L[0] + L[1] x + L[2] x^2 + ... + L[d-1] ^(d-1) of R/(f), where x stands for the image of x under the natural quotient map R -> R/(f).

If the first argument is a field_by_conway(p,n), defined in the CD finfield1, then the same interpretation holds, where R and f are respectively poly_ring_d(GFp(p),1) and conway_polynomial(p,n).

Commented Mathematical property (CMP):
later
Example:
The element x+1 of the Conway field of order 4:
$\mathrm{field_by_poly_vector}\left(\mathrm{field_by_conway}\left({\mathbb{GF}}_{{2}^{2}}\right),\left(1,1\right)\right)$
Signatures:
sts

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