1. Homework Statement
Show that f is 2-pi periodic and analytic on the strip \vert Im(z) \vert < \eta, iff it has a Fourier expansion f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}, and that a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx. Also, there's something about the lim sup of...
1. Homework Statement
Show that G is isomorphic to the Galois group of an irreducible polynomial of degree d iff is has a subgroup H of index d such that \bigcap_{\sigma \in G} \sigma H \sigma^{-1} = {1} .
2. Homework Equations
3. The Attempt at a Solution
I know that if G acts...
1. Homework Statement
I'm trying to come up with an example of a quartic polynomial over a field F which has a root in F, but whose splitting field isn't the same as its resolvent cubic.
2. Homework Equations
3. The Attempt at a Solution
Well, I know the splitting field of the...
1. Homework Statement
Let p(z) be a polynomial of degree n \geq 1. Show that \left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most n+1.
2. Homework Equations
A region (connected, open set) considered as a set in the complex plane has...
1. Homework Statement
Find the Galois closure of the field \mathbb{Q}(\alpha) over \mathbb{Q}, where \alpha = \sqrt{1 + \sqrt{2}}.
2. Homework Equations
Um...the Galois closure of E over F, where E is a finite separable extension is a Galois extension of F containing E which is minimal...
1. Homework Statement
Okay, I'm trying to explicitly determine the Galois group of x^p - 2, for p a prime.
2. Homework Equations
3. The Attempt at a Solution
Okay, so what I've come up with is that I'm going to have extensions \textbf{Q} \subset \textbf{Q}(\zeta) \subset...
1. Homework Statement
I need to show that, given F \subset E \subset K \subset L (K/F is Galois but I don't know how important that is for the part of the problem I'm having trouble with) and a homomorphism \phi:E \rightarrow L that's the identity on F, that \phi(E) \subset K.
Edit: Yeah, if...
1. Homework Statement
\int_{|z| = 2} \sqrt{z^2 - 1}
2. Homework Equations
\sqrt{z^2 - 1} = e^{\frac{1}{2} log(z+1) + \frac{1}{2} log(z - 1)}
3. The Attempt at a Solution
Honestly, my only thoughts are expanding this as some hideous Taylor series and integrating term by term...
1. Homework Statement
For prime p, nonzero a \in \bold{F}_p, prove that q(x) = x^p - x + a is irreducible over \bold{F}_p.
2. Homework Equations
3. The Attempt at a Solution
It's pretty clear that none of the elements of \bold{F}_p are roots of this polynomial. Anyway, so...
1. Homework Statement
Show that \frac{z}{(z-1)(z-2)(z+1)} has an analytic antiderivative in \{z \in \bold{C}:|z|>2\}. Does the same function with z^2 replacing z (EDIT: I mean replacing the z in the numerator, not everywhere) have an analytic antiderivative in that region?
2. Homework...
1. Homework Statement
Given an abelian group G with generators x and y, and relations 30x + 105y = 42x + 70y = 0, show it's cyclic and give its order.
2. Homework Equations
3. The Attempt at a Solution
I'm guessing the proof basically involves cleverly adding 0 to 0 to show that...
1. Homework Statement
Okay, so this is a three-part question, and I need some help with it.
1. I need to show that the function f(x) = e^{-1/x^{2}}, x > 0 and 0 otherwise is infinitely differentiable at x = 0.
2. I need to find a function from R to [0,1] that's 0 for x \leq 0 and 1 for...
I was just wondering if anyone had a decent web site explaining some of the basic terminology of differential topology. Specifically, I'm having a bit of trouble understanding charts and atlases and how one defines a smooth manifold in an arbitrary setting (i.e., not necessarily embedded in R^n)
I was wondering about the classification of groups with a certain number of subgroups. I (sort of mostly I think maybe) get the ideas behind classification of groups of a certain (hopefully small) order, but I came across a question about classifying all groups with exactly 4 subgroups, and I...
1. Homework Statement
All right, so this appeared on my final. The intervals are in the reals:
If f : [a, b] -> [c, d] , and the graph of f is closed, is f continuous?
2. Homework Equations
3. The Attempt at a Solution
Well, my gut reaction is no, just because it seems like...
1. Homework Statement
Essentially the problem is to show that a certain finite group (specifically the special linear group of order 2 over the finite field of 3 elements) is generated by two elements. But the problem is that it's non-Abelian, so I can't just consider powers of the...
Forgive the lack of TeX. I'm too lazy to type it out right now. For reference, these are problems 13 and 14 in chapter 11 of Rudin's Principles of Mathematical Analysis, slightly reworded for lack of pretty mathematical symbols.
1. Homework Statement
Problem one: I need to show that...
1. Homework Statement
For reference, this is chapter 11, problem 12 of Rudin's Principals of Mathematical Analysis.
Suppose |f(x,y)| \leq 1 if 0 \leq x \leq 1, 0 \leq y \leq 1 ; for fixed x, f(x,y) is a continuous function of y; for fixed y, f(x,y) is a continuous function of x.
Put g(x)...