# 07-401/Class Notes for March 7

## Class Plan

Some discussion of the term test and HW6.

Some discussion of our general plan.

Lecture notes

### 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 $F(a)\cong F[x]/\langle p\rangle$, 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

(This section was not covered on March 7, parts of it will be covered later on).

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.

Lemma. The property of "being relatively prime" is preserved under extensions.

Theorem. Let $f\in F[x]$ be irreducible. If $\operatorname{char}F=0$, then $f$ has no multiple zeros in any extension of $F$. If $\operatorname{char}F=p>0$, then $f$ has multiple zeros (in some extension) iff it is of the form $g(x^p)$ for some $g\in F[x]$.

Definition. A perfect field.

Theorem. A finite field is perfect.

Theorem. An irreducible polynomial over a perfect field has no multiple zeros (in any extension).

Theorem. Let $f\in F[x]$ be irreducible and let $E$ be the splitting field of $f$ over $F$. Then in $E$ all zeros of $f$ have the same multiplicity.

Corollary. $f$ as above must have the form $a(x-a_1)^n\cdots(x-a_k)^n$ for some $a\in F$ and $a_1,\ldots,a_k\in E$.

Example. $x^2-t\in{\mathbb Z}_2(t)[x]$ is irreducible and has a single zero of multiplicity 2 within its splitting field over ${\mathbb Z}_2(t)[x]$.

## Lecture Notes

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07-401 March 7 NOTES

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07-401 March 7 NOTES

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