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