189-570A: Higher Algebra I

Assignment 5

Due: Monday, November 25.

1. Show that any maximal ideal of the polynomial ring $k[x_1,\ldots,x_n]$ can be generated by $n$ linear polynomials, when $k$ is algebraically closed. Show that, when $k$ is not necessarily algebraically closed, such a maximal ideal can still be generated by at most $n$ elements.

2. If $I$ is an ideal in the polynomial ring $\mathbb C[x_1,\ldots, x_n]$, show that the radical of $I$ is the intersection of the maximal ideals that contain $I$.

3. Show that a principal ideal domain is a Dedekind domain.

4. Let $m$ be a square free integer of the form $1+4k$. Show that the ring $\mathbb Z[\sqrt{m}]$ is not integrally closed -- and hence is not a principal ideal domain, by the result of exercise 3. Give an example of a non-principal ideal in this ring.

5. Give an example of a non-Noetherian integral domain that is integrally closed in its fraction field, and in which every non-zero prime ideal is maximal.

6. Give an example of a Noetherian integral domain that is integrally closed in its field of fractions, having non-zero prime ideals that are not maximal. (The latter property implies that it is of Krull dimension $>1$.)

7. Compute the discriminant of the ring $R = \mathbb Z[\alpha]$, where $\alpha$ satisfies the minimal polynomial $x^3-x+1=0$. Use this to conclude that $R$ is a Dedekind domain. Show that $2R$ is a prime ideal of $R$, but that $5R$ and $7R$ are not. Express $5R$ and $7R$ as a product of prime ideals of $R$.

8. Show that $\mathbb Z[\sqrt{3}]$ and $\mathbb Z[\sqrt{7}]$ are the rings of integers of their respective fraction fields, but that $\mathbb Z[\sqrt{3},\sqrt{7}]$ is not integrally closed.

9.

(a) Let $R$ be principal ideal domain for which $R^\times = \pm 1$. Show that if $a$ and $b\in R$ are relatively prime and if $ab$ is the cube of an element of $R$, then $a$ and $b$ are each cubes in $R$.

(b) Let $(x,y)\in \mathbb Z^2$ be a solution to the diophantine equation $y^2 = x^3 -2$. Show that the elements $y+\sqrt{-2}$ and $y-\sqrt{-2}$ are relatively prime to each other in the ring $\mathbb Z[\sqrt{-2}]$. Assuming that $\mathbb Z[\sqrt{-2}]$ is a unique factorisation domain, use this along with the result obtained in (a) to give a complete list of integer solutions of the equation $y^2=x^3-2$.

(c) The proof sketched above was obtained by Euler, who took it for granted that $\mathbb Z[\sqrt{-2}]$ is a unique factorisation ring. Complete Euler's argument by showing that $\mathbb Z[\sqrt{-2}]$ is a Euclidean domain, and therefore has unique factorisation.

10.

(a) By following the same line of reasoning as Euler's in question 9, show that if the ring $\mathbb Z[\sqrt{-334}]$ is a unique factorisation ring, then the Diophantine equation $y^2 = x^3-334$ has no integer solutions.

(b) Note, however, that $(x,y) = (7,3)$ is in fact a solution to this equation, and conclude that $\mathbb Z[\sqrt{-334}]$ must in fact fail to satisfy unique factorisation. Can you parlay the solution $(7,3)$ into a counterexample to unique factorisation in this ring?