07-401/Class Notes for March 7

07-401/Navigation Panel   # Week of... Links 1 Jan 10 About, Notes, HW1 2 Jan 17 HW2, Notes 3 Jan 24 HW3, Photo, Notes 4 Jan 31 HW4, Notes 5 Feb 7 HW5, Notes 6 Feb 14 On TT, Notes R Feb 21 Reading week 7 Feb 28 Term Test 8 Mar 7 HW6, Notes 9 Mar 14 HW7, Notes 10 Mar 21 HW8, E8, Notes 11 Mar 28 HW9, Notes 12 Apr 4 HW10, Notes 13 Apr 11 Notes, PM S Apr 16-20 Study Period F Apr 24 Final Add your name / see who's in! Register of Good Deeds

In Preparation

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

Class Plan

Some discussion of the term test and HW6.

Some discussion of our general plan.

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.

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 ${\mathbb {Z} }_{2}(t)[x]$.