Recent content by masterslave

Homework Statement The problem is from Sarason, page 44, Exercise IV.14.1. Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function. Assume that f(G) is open, that g is continuous, and that f\prime\neq 0\forall z\in G. Prove g is...

Homework Statement The problem, for reference, is from Sarason's book "Complex Function Theory, 2nd edition" and is on page 81, Exercise VII.5.1. Let C be a counterclockwise oriented circle, and let f be a holomorphic function defined in an open set containing C and its interior. What is...

I see all of it, except the logic you use to conclude that \alpha is a primitive root of unity. My thought is because p(x) is irreducible and \alpha, \alpha^2, \alpha^3, \ldots are all roots of it, then \alpha =\alpha^2 =\alpha^3 =\ldots =1 implying that \alpha is a root of unity. Any...

Wow, yeah, it's so obvious now. Thank you all for your time. I get it now.

Homework Statement Let p\in\mathbb{Q}[x] be an irreducible polynomial. Suppose K is an extension field of \mathbb{Q} that contains a root \alpha of p such that p(\alpha^2)=0. Prove that p splits in K[x]. The Attempt at a Solution I was thinking contradiction, but if p does not split in...

Thank you both for your kind help. I see how \sigma(E) \subseteq L , but i don't see how to get L \subseteq \sigma(E) Thanks again!

Homework Statement Let K be a Galois extension of F. Two intermediate fields E and L of field F are said to be conjugate if there exists \sigma\in\text Gal_F K such that \sigma (E) = L. Prove that E and L are conjugates of F if and only if \text Gal_E K and \text Gal_L K are conjugate...

It's an integration by part question. First, use a substitution to get it to one variable instead of a polynomial in the logarithm. w=t+1 dw=dt Substitute (I'm going to revert the limits in the end to the original variable, so you know): Integrate of ln(w)dw Let: u=ln(w) and dv=dw du=(1/w)*dw...