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% \title{Math 210B: Homework 1}
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\centerline{\Large Math 210B: Homework 1}

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Readings in Lang's {\textit{Algebra}} for these problems: Sections 7.1 and 8.1;
also Section 4.2 for Gauss' Lemma and Section 6.5 for norm and trace.

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For the first two problems, let $R$ be an integral domain, $F$ its field of
fractions, $K / F$ a larger field. Let $\alpha \in K$. As we recall, $\alpha$
is defined to be integral over $R$ if it satisfies a monic polynomial $f
(\alpha) = 0$ where
\[ f (x) = x^n + a_{n - 1} x^{n - 1} + \ldots + a_0 \in R [x] . \]
However in this definition we do not require $f$ to be irreducible. So
$\alpha$ is also a root of a monic polynomial
\[ g (x) = x^m + b_{m - 1} x^{m - 1} + \ldots + b_0 \in F [x] \]
that is irreducible in $F [x]$. The first two exercises give conditions for $g
\in R [x]$.


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\textbf{Problem} 1. Suppose that $R$ is a unique factorization domain. Use Gauss' Lemma to
show that $g (x) \in R [x]$.

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The next problem requires some Galois theory.

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\textbf{Problem} 2. Now suppose that $R$ is an integrally closed integral
domain and $\operatorname{char} (F) = 0$. Let
$\alpha_1, \cdots, \alpha_m$ be the roots of $g$ in a splitting field
$E \supseteq K$. Thus $m=\deg(g)$.
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(a) Explain why the $\alpha_i$ are distinct and
\[ g (x) = (x - \alpha_1) \cdots (x - \alpha_m) . \]
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(b) Prove that $b_i \in R$, so just as in Problem~1, $g \in R [x]$.

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\textbf{Problem} 3. Let $D$ be squarefree and consider $\alpha = a + b \sqrt{D} \in \mathbb{Q}
\left( \sqrt{D} \right)$. Describe the norm and trace $\operatorname{tr} (\alpha)$ and
$N (\alpha)$. Prove that $\alpha$ is integral over $\mathbb{Z}$ if and only if
$\operatorname{tr} (\alpha)$ and $N (\alpha)$ are in~$\mathbb{Z}$.

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\textbf{Problem} 4. Let $p$ be a prime. Determine the integral closure of $\mathbb{Z}$ in
$\mathbb{Q} \left( \sqrt{p} \right)$ and $\mathbb{Q} \left( \sqrt{- p}
\right)$. The shape of the answer should depend on $p$ mod $4$.

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\textbf{Problem} 5. Let $E \supset K \supset F$ fields. Prove that the transcendence degrees
are additive:
\[ \operatorname{tr} . \deg (E / F) = \operatorname{tr} . \deg (E / K) + \operatorname{tr} . \deg (K /
   F) . \]

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