189-570A: Higher Algebra I

Take-Home Final: Frequently Asked Questions

From Lorna McNair: I am experiencing a few difficulties with the question involving the differential galois theory paper. The question asks us to supply all the details of the proof but I dont know what a few of the terms mean. The terms in question are " two-step" " Integral Heisenberg group" and "unipotent".

The integral Heisenberg group is the group of 3 x 3 upper-triangular matrices with integer entries and ones on the diagonal. A unipotent element is a matrix all of whose eigenvalues are equal to one, and a unipotent group is a grop consisting only of such elements. Finally, a two-step group is a solvable group whose commutator subgroup is abelian.

From Austin Roche: In 1(b) the restriction that the characters in question be _complex_ is relaxed. So can the associated field be any field, or must it be commutative?

All fields are assumed to be commutative, unless it is otherwise explicitly mentionned. Also, the field in question 1(b) is assumed to be of characteristic zero.

From Ramon Casanova: A question about problem 2a) on the take-home exam. It says that if f(x) in Z[x] has a root modulo every prime p, then it has a root in Q. I remembered from number theory that either 2,3, or 6 must be a quadratic residue modulo every prime p, so the polynomial: (x^2 - 2)(x^2 - 3)(x^2 - 6) must have a root modulo every prime p, but this has no rational roots. I was wondering if perhaps I was missing something or if the question needs to be changed a bit.

This is completely true. So I goofed! What I REALLY wanted to ask was the following: Show that an irreducible poynomial cannot have a root mod p for all p.

From Amelie Schinck: I was just wondering if we can take for granted in the exam that GL_2(F_p) and GL_3(F_2) are simple groups? (since it has been seen in previous algebra courses and it takes a while to prove it).

Actually, you can show this without too much effort using what we did in class, so I appreciate if a few lines of proof are included. You always get extra brownie points for short, elegant proofs as opposed to long exhausting ones, in questions of this sort. But be careful, by the way: GL₂(F_p) is not simple!

From Sidney Trudeau: On the differential galois theory stuff, although no specific mention is made in the paragraph, I assume D = d/dt ?

Yes, that's right.

From alot of different people in the class: There seems to be some uncertainty about what is being asked of you in the question on differential Galois theory. This question should be taken in the spirit of an essay question. The goal is for you to convince me that you've read the article and understood enough of it to be able to carry out some some of the calculations (those that are claimed to be "easy verifications" in that paragraph). With what you have learned of Galois theory in the course, you are in a situation where you should be able to take a text like this, read it, and understand it. Reading the mathematics literature, and being able to extract the essential meaning of a text even if you have not been introduced formally to all the words and concepts, is an important skill that comes in very useful when writing a thesis, just as much as the problem-solving skills that are usually emphasized on mathematics exams!

From Lorna McNair comes the question about what "Zariski closure" means. The Zariski topology on GL_n(C) is the topology in which the closed sets are defined to be the zero sets of polynomial equations with complex coefficients. The Zariski closure of a set is then the smallest Zariski-closed set which contains it.