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189-571B: Higher Algebra II
Assignment 1. Due: Wednesday, February 20.
1. Let $p$ and $q$ be two primes that are congruent to $3$ modulo $4$.
Show that $\mathbb Z[\sqrt{p}]$ and $\mathbb Z[\sqrt{q}]$ are the rings of integers of $\mathbb Q(\sqrt{p})$ and
$\mathbb Q(\sqrt{q})$ respectively, but that
$\mathbb Z[\sqrt{p},\sqrt{q}]$ is not the ring of integers of $\mathbb Q(\sqrt{p},\sqrt{q})$.
Compute the ring of integers of this biquadratic field.
2.
Let $k$ be a field and let $A=k[x^2,x^3]$ be the subring of the polynomial
ring $k[x]$ generated by $x^2$ and $x^3$.
Show that $A$ is Noetherian and of Krull dimension one, but that $A$ is not integrally closed.
3.
Let $p$ be a prime and let $K=\mathbb Q(\zeta)$ be the field generated over $\mathbb Q$ by
a primitive $p$-th root of unity $\zeta$.
(a) Show that the ring $\mathcal O_K :=
\mathbb Z[\zeta]$ is the ring of integers of $K$.
(b) For each rational prime $q$ of $\mathbb Z$, show that the ideal $q\mathcal O_K$
of $\mathcal O_K$ factors as
$q\mathcal O_K = (q_1 \cdots q_t)^e$,
where each $q_i$ is a prime ideal of residue degree $f$, and $tef=p-1$.
(c) Give an explicit formula for $t$, $f$ and $e$ in terms of $q$ (and $p$, of course).
4.
Kunz, exercise 4, page 9.
5.
Kunz, exercise 5, page 9.
6.
Kunz, exercise 1, page 15.
7.
Kunz, exercise 6, page 15.
8.
Kunz, exercise 9, page 16.
9.
Kunz, exercise 1, page 21.
10.
Kunz, exercise 5, page 22.