Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates by Robert B. Ash

Author:Robert B. Ash
Language: eng
Format: epub, pdf
Publisher: Dover Publications
Published: 2000-06-14T16:00:00+00:00

where r = [E : F(x)].

Proof. First assume that r = 1, so that E = F(x). By the Cayley-Hamilton theorem, the linear transformation m(x) satisfies char(x), and since m(x) is multiplication by x, x itself is a root of char(x). Thus min(x, F) divides char(x). But both polynomials have degree n, and the result follows. In the general case, let y1, …, ys be a basis for F(x) over F, and let z1, …, zr be a basis for E over F(x). Then the yizj form a basis for E over F. Let A = A(x) be the matrix representing multiplication by x in the extension F(x)/F, so that . Order the basis for E/F as y1z1, y2z1, …, ysz1; y1z2, y2z2, …, ysz2; … y1zr, y2zr, …, yszr. Then m(x) is represented in E/F as

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