Degree - Maple Help

Algebraic

 Degree
 formal degree of an algebraic extension

 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

 > $\mathrm{Algebraic}\left[\mathrm{Degree}\right]\left(\left\{\mathrm{RootOf}\left({x}^{2}-2,\mathrm{index}=1\right),\mathrm{RootOf}\left({y}^{2}-\mathrm{RootOf}\left({x}^{2}-2,\mathrm{index}=1\right),\mathrm{index}=1\right)\right\}\right)$
 ${4}$ (1)
 > $\mathrm{Algebraic}\left[\mathrm{Degree}\right]\left(\varnothing \right)$
 ${1}$ (2)

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

 > $\mathrm{Algebraic}\left[\mathrm{Degree}\right]\left(\left\{\mathrm{RootOf}\left({x}^{2}-2,\mathrm{index}=1\right),\mathrm{RootOf}\left({x}^{2}-3,\mathrm{index}=1\right),\mathrm{RootOf}\left({x}^{2}-6,\mathrm{index}=1\right)\right\}\right)$
 ${8}$ (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 $4$:

 > $\mathrm{Algebraic}\left[\mathrm{Degree}\right]\left(\left\{\mathrm{RootOf}\left({y}^{2}-\mathrm{RootOf}\left({x}^{2}-2,\mathrm{index}=1\right),\mathrm{index}=1\right)\right\}\right)$
 ${2}$ (4)