07-401/Class Notes for March 7

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In Preparation

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

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 $p$.

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 ${\mathbb {Z} }_{3}[x]/\langle x^{2}+x+2\rangle$.

Theorem. Any two splitting fields for $f\in F[x]$ over $F$ are isomorphic.

Lemma 1. 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 $F[a]\cong F'[a']$.

Lemma 2. Isomorphisms can be extended to splitting fields.

Zeros of Irreducible Polynomials

Definition. The derivative of a polynomial.

Claim. The derivative operation is linear and satisfies Leibnitz's law.

Theorem. $f\in F[x]$ has a multiple zero in some extension field of $F$ iff $f$ and $f'$ have a common factor of positive degree.