It seems like you're going about the problem correctly except that obviously the splitting field won't be as described if the quadratic is reducible. Other than that, unless I'm forgetting something (which would hardly be surprising), what you're saying is completely true.
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...
Well, it looks like you're trying to find the power series of ln(5 - x) by differentiating the series for 1/(5 - x) term by term. But ln(5 - x) is the integral of 1/(5 - x) (give or take a sign).
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} .
Homework Equations
The Attempt at a Solution
I know that if G acts transitively as a...
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.
Homework Equations
The Attempt at a Solution
Well, I know the splitting field of the cubic...