(2) Let K | F be a field extension and A € M₁ (F). Denote its minimal polynomial by A,F, and denote it by A,K if we consider A as an element of Mn(K). From the definition of minimal polynomials it's clear that μA,K divides A,F in K[x]. Explain why here (as opposed to the situation for mini- mal polynomials of elements of field extensions) we always have μA,K = μA,F. Remark: There are multiple ways to approach this question.

(2) Let K | F be a field extension and A € M₁ (F). Denote its minimal polynomial by A,F, and denote it by A,K if we consider A as an element of Mn(K). From the definition of minimal polynomials it's clear that μA,K divides A,F in K[x]. Explain why here (as opposed to the situation for mini- mal polynomials of elements of field extensions) we always have μA,K = μA,F. Remark: There are multiple ways to approach this question.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 25E: Suppose S is a subset of an field F that contains at least two elements and satisfies both of the...

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