# Tag Info

## Hot answers tagged extension-field

Accepted

### Changing the field of an irreducible representation leaves it irreducible.

I don’t think that’s correct, but neither do I know a counter-example (is there a representation $V$ of a group $G$ that can be defined on two number fields $X,Y$ that are linearly disjoint with ...
• 35.2k
Accepted

### On Frobenius–Schur indicator of real/complex representations

It means both in different contexts, as far as I know. It's unfortunate but they can be disambiguated based on whether the author is discussing real or complex representations. The Frobenius-Schur ...
• 436k
1 vote

### Is dependent choice what one must use in this step of Artin's construction of the algebraic closure?

This is not an answer to the question you asked, but three comments: First, note that in the construction of $K_1$ alone you already need the axiom of choice just to choose the maximal ideal $m$. ...
• 436k
1 vote

### Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

The answer to your first question is yes, $F_{\infty}$ has only one subfield of degree $2^n$. Indeed, writing $F_n = \mathbb{Q}(2^{\frac{1}{2^n}})$, you have $F_n \subset F_m$ whenever $n\leq m$. ...
1 vote
Accepted

### Defining polynomial for a compositum of splitting fields

If the resultant is separable, it should be irreducible precisely when $L_i \cap L_j = K$. This follows from a general result about composita of field extensions (e.g., Proposition 21 in Section 13.2 ...
• 370
1 vote
Accepted

### Natural way to extend the ring $\mathbb{Z} / p^k \mathbb{Z}$ so that the equation $x^2 + 1 \equiv 0 (\text{mod }p^k)$ has a solution

A simple way to define your ring is to start with the ring of Gaussian integers and take the congruence modulo $p^k$ on this ring. Or directly, consider the set  R = \bigl\{x + iy \mid x, y \in {\...
• 40.8k
1 vote

### Inertia field example of $\mathbb{Q}_5(\sqrt[4]{50})$

Have you gotten through to full understanding yet? The comment of @user8268 gives the answer, but without explanation. Here’s how I look at things. Your extension is quartic, so you might as well ...
• 63.4k

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