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\begin{document}
\title{Math 200b Winter 2021 Homework 6}
\date{Due 2/26/2020 by midnight on Gradescope}
\maketitle


1.  Let $F$ be a field of characteristic not $2$.   Let $F \subseteq K$ be a field extension, and let $a, b \in F$ be elements, neither 
of which is a square in $F$.  Let $\sqrt{a} , \sqrt{b} \in K$ be roots of the polynomials $x^2 - a, x^2 - b \in F[x]$, respectively.   Prove 
that $[F(\sqrt{a}, \sqrt{b}): F] = 4$ if and only if $ab$ is not a square in $F$, and $[F(\sqrt{a}, \sqrt{b}): F] = 2$ otherwise.   

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2(a).  Let $F \subseteq K$ be a field extension.  Suppose that $F \subseteq K_1 \subseteq K$ and $F \subseteq K_2 \subseteq K$ where 
$K_1$ and $K_2$ are subfields of $K$.  The composite field $K_1 K_2$ is defined to be the smallest subfield of $K$ containing both $K_1$ and $K_2$.  Show that if $[K_1:F] < \infty$ and $[K_2 :F] < \infty$ then 
$K_1 K_2$ can also be described as the usual notation for products of subsets of a ring suggests:
\[
K_1 K_2 = \bigg\{ \sum_{i=1}^d a_i b_i \bigg| a_i \in K_1, b_i \in K_2, d \geq 0  \bigg\} \subseteq K.
\]
(Hint: show that $K_1 K_2$ is a subring of $K$--- then why is it forced to be a subfield?)

(b).   Show that  if $[K_1:F] < \infty$ and $[K_2 :F] < \infty$ as in (b), then there is a surjective homomorphism of $F$-algebras $\theta: K_1 \otimes_F K_2 \to K_1 K_2$ given by $\theta(a \otimes b) = ab$.    Prove that $\theta$ is an isomorphism if and only if $[K_1K_2: F] = [K_1:F][K_2:F]$.  

(c).  Show that the $\mb{Q}$-algebra $\mb{Q}(\sqrt{2}) \otimes_{\mb{Q}} \mb{Q}(\sqrt{3})$ is a field which is isomorphic to $\mb{Q}(\sqrt{2}, \sqrt{3})$.

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3. Suppose that $f \in F[x]$ has degree $n$ 
and that $K$ is a splitting field for $f$ over $F$.  
Show that $[K:F] \leq n!$.

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4.  Let $K$ be the splitting field of $x^6 -4$ over $\mb{Q}$.  Find $[K: \mb{Q}]$.

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5.  Let $\alpha = \sqrt{ 2 + \sqrt{2}} \in \mb{C}$ and let $f$ be the minimal polynomial of $\alpha$ over $\mb{Q}$.

(a).  Compute $f$.

(b).  Let $K$ be a splitting field for $f$ over $\mb{Q}$.  Find $[K:\mb{Q}]$.

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6.  Let $K$ be a field with $\operatorname{char} K = p > 0$.  Define $K^p = \{ a^p | a \in K \}$ and recall that $K$ is called \emph{perfect} if $K^p = K$.  In this problem 
let $K$ be a nonperfect field.  

(a).  Consider the extension $F = K^p \subseteq K$.  
Show that for every $\alpha \in K - F$, $\operatorname{minpoly}_{F}(\alpha)$ is an inseparable polynomial of degree $p$.

(b).  Show that if $[K: F] > p$ then the extension $K/F$ has no primitive element, i.e. there is no $\gamma \in K$ such that $K = F(\gamma)$.

(c).  Give an explicit example where $[K:F] < \infty$ and the situation of part (b) occurs; so finite degree extensions in characteristic $p$ do not always have primitive elements. 









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