07-401/Class Notes for March 7

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Contents

Class Plan

Some discussion of the term test and HW6.

Extension Fields

Definition. An extension field E of F.

Theorem. For every non-constant polynomial f in F[x] there is an extension E of F in which f has a zero.

Example x^2+1 over {\mathbb R}.

Example x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2) over {\mathbb Z}/3.

Definition. F(a_1,\ldots,a_n).

Theorem. If a is a root of an irreducible polynomial p\in F[x], within some extension field E of F, then F(a)\cong F[a]/\langle p\rangle, and \{1,a,a^2,\ldots,a^{n-1}\} (here n=\deg p) is a basis for F(a) over F.

Corollary. In this case, F(a) depends only on p.

Corollary. If p\in F[x] irreducible over F, \phi:F\to F' an isomorphism, a a root of p (in some E/F), a' a root of \phi(p) in some E'/F', then F[a]\cong F'[a'].

Splitting Fields

Definition. f\in F[x] splits in E/F, a splitting field for f over F.

Theorem. A splitting field always exists.

Example. x^4-x^2-2=(x^2-2)(x^2+1) over {\mathbb Q}.

Example. Factor x^2+x+2\in{\mathbb Z}_3[x] within its splitting field {\mathbb Z}_3[x]/\langle x^2+x+2\rangle.

Zeros of Irreducible Polynomials

Perfect Fields