Homework 1

1. Find the radical of the ideal $(x^3-y^6, xy-y^3)$ in $\mathbb{C}[x,y]$.
2. Determine generators of the ideal of $X$, where $X$ is the union of the three coordinate axes in $\mathbb{A}^3$.  Prove that you need at least 3. 
3. Show that every non-empty open subset of an irreducible topological space is dense.  If $Y$ is a subset of a topological space $X$, which is irreducible in its induced topology, then the closure $\overline{Y}$ is also irreducible. 
4. Find the irreducible components of $Z(x^2-yz,xz-x)\subset \mathbb{A}^3$. 
5. Let $f$ be an irreducible polynomial of degree 2 in $k[x,y]$, and let $X=Z(f)$ be the conic defined by $f$.  Then $X$ is either isomorphic to the parabola $Z(y-x^2)$ or the hyperbola $Z(xy-1)$, but not both. 

Please write your solutions on paper and scan/photograph, or prepare a pdf from a typed or (la)texed file and submit online via Canvas before the deadline.  

Collaboration is fine, in fact encouraged,  but please write your own solutions. 

As part of the assignment you will be expected to read the solutions of two of your peers and comment on them.  (This is process is anonymous, don't write your name on your solution.)

Remember that you are writing for a human to understand your solutions.  It is more important to explain the main ideas entering the proof, than to write a formal proof that would compile on a computer. In fact, your solutions will primarily be read be your peers (other students in this class), so write your solutions to explain to your peers why or how your proof works.  I (your instructor) will read some but not all of your solutions (I'm sorry).  Your grade will be some combination of the number of problem solutions you submitted, and the quality of the feedback you gave to your peers as part of the peer review process.

It is hard to find errors or gaps in homework solutions, both for me, and for your peers.  So if you are not sure that your proof is sound, it is better to ask about potential gaps, than to assume it's correct if no one found a mistake.

When you are reviewing your peers' solutions, please be kind.  Point out if you think there are gaps, or an argument does not convince you.  Give praise for elegant solutions!