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189-571B: Higher Algebra II
Assignment 1. Due: Wednesday, January 30.
1. Let $k$ be an algebraically closed
field of characteristic $0$. Show that the curve with equation $y^2 = f(x)$, where $f(x)\in k[x]$ is a non-zero polynomial, is non-singular if and only if the ring $k[x,y]/(y^2-f(x))$ is integrally closed over $k[x]$.
(Remark: the assertion continues to hold when $k$ is not algebraically closed, provided one is attentive to the fact that the statements
``$X$ is a non-singular curve over $k$" means something very different
from ``the set $X(k)$ of $k$-rational points on $X$ contains no singular points". It is worth spending a bit of time to ponder the difference.)
2. Which of the following rings are Dedekind domains?
Justify your answers.
(a) $R= {\bf Z}[x]/(x^2+9)$;
(b) $R= {\bf Z}[x]/(9x^2+1)$;
(c) $R= F[x,y]$ where $F$ is a field.
(d) $R = F[x,y]/(y^2-x^3-1)$, where $F$ is a field.
3.
Problem 2-2, page 44 of
Milne, ANT.
4.
Problem 2-4, page 44 of
Milne, ANT.
5.
Problem 2-6, page 44 of
Milne, ANT.
6.
Problem 2-7, page 44 of
Milne, ANT.
7.
Problem 2-8, page 44 of
Milne, ANT.
8. Compute the ring of integers of
the field ${\mathbb Q}(2^{1/3})$.
9. A Dedekind domain
is a domain which (a) is Noetherian, (b) is integrally closed and (c) in which every non-zero prime ideal is maximal.
For each of (a), (b) and (c), give an example of a ring that does not satisfy that property but satisfies the other two.
10. Examine to what extent unique factorisation (into irreducibles, or into prime ideals) holds in the rings that you have produced in
exercise 9.