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Section Day 18
This is an outline of the topics we covered in the eighteenth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Tuesday 7/21
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State the following mathematical results: Gauss’s Lemma, Reducibility over Q implies Reducibility over Z
State and apply the following irreducibility tests: Mod
\(p\) Irreducibility Test, Eisenstein’s Criterion
State the following mathematical results:
\(\ideal{p}\) is Maximal Iff
\(p\) is Irreducible and Corollary
Algebraist of the Day.
Christine Kelley
Algebraist and coding theorist at University of Nebraska Lincoln
Current director of MAA Project NExT, Co-PI for the Nebraska Conference for Undergraduate Women in Mathematics
Theorem 183 . F[x] is a PID.
Let
\(F\) be a field. Then
\(F[x]\) is a principal ideal domain.
Proof.
Corollary 184 . Ideal Generators Have Minimal Degree in \(F[x]\) .
Let
\(F\) be a field,
\(I\) be a nonzero ideal in
\(F[x]\text{,}\) and
\(g \in I\text{.}\) Then
\(I=\ideal{g}\) if and only if
\(g\) is a polynomial of minimal degree in
\(I\text{.}\)
Theorem 185 . Reducibility Over \(\Q\) Implies Reducibility Over \(\Z\) .
Let
\(f(x)\in \Z[x]\text{.}\) If
\(f(x)\) is reducible over
\(\Q\text{,}\) then it is reducible over
\(\Z\text{.}\)
Proof.
Theorem 186 . Mod \(p\) Irreducibility Test.
Let
\(p\) be a prime and suppose that
\(f(x)\in\Z[x]\) with
\(\deg f(x)\geq 1\text{.}\) Let
\(\bar{f}(x)\) be the polynomial in
\(\Z_p[x]\) obtained by reducing the coefficients of
\(f(x)\) modulo
\(p\text{.}\) If
\(\bar{f}(x)\) is irreducible over
\(\Z_p\) and
\(\deg \bar{f}(x)=\deg f(x)\text{,}\) then
\(f(x)\) is irreducible in
\(\Q[x]\text{.}\)
Proof.
Example 187 . Applying the Mod \(p\) Test.
Theorem 188 . Eisenstein’s Criterion.
Let
\begin{equation*}
f(x)=a_n x^n + a_{n-1} x^{n-1}+\dots + a_0 \in \Z[x]\text{.}
\end{equation*}
If there is a prime \(p\) such that \(p \nmid a_n, p \mid a_{n-1}, \dots, p\mid a_0\) and \(p^2 \nmid a_0\text{,}\) then \(f(x)\) is irreducible over \(\Q\text{.}\)
Example 189 . Applying Eisenstein’s Criterion.
Why do we care about irreducibility?
Theorem 190 . Irreducible Polynomials Generate Maximal Ideals.
Let
\(F\) be a field and
\(p(x) \in F[x]\text{.}\) Then
\(\ideal{p(x)}\) is maximal in
\(F[x]\) if and only if
\(p(x)\) is irreducible over
\(F\text{.}\)
Proof.
Corollary 191 . New Fields from Old Fields.
Let
\(F\) be a field and
\(p(x)\) be irreducible over
\(F\text{.}\) Then
\(F[x]/\ideal{p(x)}\) is a field.
