Search Results

Another trick, using the field trace, is the described in exercise 16 of Chapter 2 of Marcus's Number Fields, though as far as I can tell this only works for the special case of radicals (since their traces … because I recently had the following messy situation: $K$ is the splitting field of a cubic $g\in\mathbb{Q}(t)[x]$, having $[K:\mathbb{Q}(t)]=3$, and $f_1$, $f_2\in K[x]$ are two cubics with splitting fields …

However, this doesn't exist even for number fields, e.g. setting $K=\mathbb{Q}$, $L=\mathbb{Q}(\zeta_3,\sqrt[3]{2})$, $E_1=\mathbb{Q}(\sqrt[3]{2})$, and $E_2=\mathbb{Q}(\zeta_3\sqrt[3]{2})$, we have $\ … So, to prove the claim / construct a counterexample, it seems to me that we want to look at intermediate fields $E$ which are maximal among those such that $\mu_E=\mu_K$, and determine whether or not there …

I was feeling a bit rusty on my field theory, and I was reviewing out of McCarthy's excellent book, Algebraic Extensions of Fields. …

This seems like an obvious fact, but I'm not sure what the necessary meaning of "nice" is to get a result like this. I'm wondering if there is a theorem of the form: For any <1> field extension $K/F …

So, equivalently, suppose we have a symmetric function $S( , , , )$ such that $S(z_1,z_2,z_3,z_4)=0$ whenever $z_1,z_2,z_3,z_4$ are conjugates over $\mathbb{Q}$ and such that $\mathbb{Q}(z_i)\supset\m …

I believe they have a unary "inverse" operation like meadows, but I assume something is different about them. neofield, which (according to this paper) appear to be fields, without the associativity of … explain which classic concepts/theorems about fields carry over to each structure (do they have a notion of algebraic elements? is there a Galois-like theory for them? …