compute subfields of an extension field - Maple Help

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Subfields - compute subfields of an extension field

Calling Sequence

Subfields(f,deg,K,x)

Parameters

f

-

polynomial or set of polynomials

deg

-

positive integer

K

-

set of RootOfs

x

-

variable

Description

• 

The Subfields function is a placeholder for representing a primitive description of an algebraic extension. It is used in conjunction with evala.

• 

Let f be an irreducible polynomial in K[x]. If f contains only one variable then x need not be specified, otherwise both K and x must be specified. If the argument K is not specified then K is the smallest extension of the rationals such that the coefficients of f are in K. If K is specified then the field K contains the RootOfs in this set as well. Let L be the field extension of K given by one single root of f. So L is not the splitting field; L = K[x]/(f) = K(RootOf(f,x). The call evala(Subfields(f, deg, K, x)) computes the set of all subfields of L over K of degree deg. Each subfield is given by a single RootOf of degree deg.

• 

A field K(R) where R is a RootOf is a subfield of L if and only if f has an irreducible factor g over K(R) such the degree of f equals the product of the degree of g and the degree of R.

• 

If f is not a polynomial but a set of polynomials then this procedure computes those subfields that the elements of f have in common. Each of these polynomials must be irreducible over K, otherwise this procedure may not work correctly.

Examples

evalaSubfieldsx4+1,2

RootOf_Z22,RootOf_Z2+1,RootOf_Z2+2

(1)

evalaSubfieldsx4+1,3

(2)

evalaSubfieldsx4+1,x4+2,2

RootOf_Z2+2

(3)

See Also

evala, RootOf


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