07-401/Class Notes for March 7

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+1} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb R}} .

Example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^5+2x^2+2x+2=(x^2+1)(x^3+2x+2)} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}/3} .

Definition. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a_1,\ldots,a_n)} .

Theorem. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is a root of an irreducible polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\in F[x]} , within some extension field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} of $F$, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a)\cong F[x]/\langle p\rangle} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,a,a^2,\ldots,a^{n-1}\}} (here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=\deg p} ) is a basis for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(a)} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} .

Corollary. In this case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} all zeros of $f$ have the same multiplicity.

Corollary. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} as above must have the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(x-a_1)^n\cdots(x-a_k)^n} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1,\ldots,a_k\in E} .

Example. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-t\in{\mathbb Z}_2(t)[x]} is irreducible and has a single zero of multiplicity 2 within its splitting field over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2(t)[x]} .

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