# Questions tagged [field-theory]

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. This tag is NOT APPROPRIATE for questions about the fields you encounter in multivariable calculus or physics. Use (vector-fields) for questions on that theme instead.

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### Does $a\neq b$ imply $a\otimes c\neq b\otimes c$ (when $c\neq 0$)?

I'm trying to understand by myself Hopf-Galois theory, and for an example to verify certain properties I need a map to be a bijection. The details of my example are not important, what I just need is ...
1 vote
42 views

### Prove that $A$ is a field [duplicate]

I've trying to prove some theorem about Galois-Hopf Theory. It's details are not necessary to understand my question. I want to prove the following: Let $F/K$ be a Galois field extension. Let $A$ be ...
10 views

### $k\subset K$ field extension, $S$ transcendence base, then $K$ is algebraic extension of $k(S)$, why?

Terminology from Lang, Chapter VIII. 1. $k\subset K$ field extension and $S$ transcendence base, then $K$ is algebraic over $k(S)$. Why is that so? Lang states as it is something trivial. Surely we ...
39 views

### Confusion regarding a proof in Galois theory.

I am studying Galois theory from Malik,Mordeson,Sen.There is a theorem which is as follows: $K/F$ be a finite extension and $\alpha\in K$,then there exists a polynomial $g(x)$ and a field $E\supset K$...
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### Why do we require uniqueness in the universal property for a fraction field?

I'm a university student taking a course in abstract algebra. My professor recently introduced fraction fields, giving this definition: Let $R$ be an integral domain. There exists a field $F$, called ...
151 views

### If $f(x)$ is irreducible, is $f(x^k)$ irreducible?

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer? ...
38 views

### Solvability of the quintic by radicals - missing step?

A theorem due to Galois asserts that a polynomial $f\in F[x]$ can be solved by radicals iff. the Galois group of $f$ is a solvable group. In my lecture notes as a corollary I have the following: The ...
24 views

### I want someone to explain example 1 [duplicate]

enter image description hereenter image description here I want someone to explain example 1 clearly
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### Are there any interpretations of field theory which would allow for a negative degree of a field extension?

I've only ever seen finite field extensions indicated as $[L:K] < \infty$. I've never seen $-\infty<[L:K]<\infty$. I take this to mean that field extensions of a negative degrees are not ...
1 vote
24 views

### Every finite separable extension is contained in a Galois extension

I am having trouble understanding the following proof: Claim: Let $K/F$ be a finite separable field extension. Then $K$ is contained in a Galois extension $K \supset L \supset F$. Proof: Since the ...
### Two seemingly different totally ramified extension,$\Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$
$\Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$ are both totally ramified extension over $\Bbb{Q}_p$ each has extension degree $p^n-p^{n-1}$ and $n$. The former can be regarded as Lubin Tate extension,...