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Algebraic

  

Degree

  

formal degree of an algebraic extension

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Degree(S)

Parameters

S

-

set of RootOfs of type algext

Description

• 

Given a set of RootOfs defining an algebraic extension, Degree determines the formal algebraic degree of that extension.

• 

The formal algebraic degree of a non-nested RootOf is the degree of its first argument with respect to the variable _Z, i.e., the degree of its defining polynomial.

• 

The formal algebraic degree of a set of RootOfs is the product of the degrees of all the defining polynomials.

• 

This procedure assumes that all sub-RootOfs of a RootOf in S are elements of S as well.

• 

If all RootOfs in S are independent, then the formal algebraic degree is equal to the actual degree of the field extension defined by the RootOfs in S over the ground field.

Examples

AlgebraicDegreeRootOfx22,index=1,RootOfy2RootOfx22,index=1,index=1

4

(1)

AlgebraicDegree

1

(2)

The RootOfs in the following example are dependent, and the formal algebraic degree is bigger than the actual degree, which is 4:

AlgebraicDegreeRootOfx22,index=1,RootOfx23,index=1,RootOfx26,index=1

8

(3)

The set of RootOfs in the following example is not closed under sub-RootOfs, and Degree does not return the correct formal degree, which is 4:

AlgebraicDegreeRootOfy2RootOfx22,index=1,index=1

2

(4)

See Also

Algebraic

degree