# 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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### Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
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### x^7-1 over F_3 is not solvable by radicals [closed]

Showing that the polynomial $x^7-1$ over $\mathbb{F}_3$ is not solvable by radicals.
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### Tower laws, intersection and product of fields

I am working on a problem sheet question of Galois theory where I get stuck: Assume all field extensions here are finite. Consider subfields $E$ and $F$ of $\Omega$. Let $EF$ denote the smallest ...
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### Show that $f(t)=t^8-2t^4+9$ is the minimal polynomial of $\alpha = \sqrt{i+\sqrt 2}$ over $\mathbb{Q}$

I am really struggling to show that. I can't find a proof for $f$ to be irreducible. Eisentsien doesn't work. Revesing doesn't lead me anywhere and mod p didn't work as well, is there any criterion I ...
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### Compute the Galois group for $f(x) = (x^2-p)(x^2-q)(x^2-pq)$ over $\mathbb{Q}$ and determine all subfields of splitting field

For $f(x) = (x^2-p)(x^2-q)(x^2-pq) \in \mathbb{Q}[x]$ where $p\neq q$ are primes I need to compute the Galois group for $f$ over $\mathbb{Q}$ and determine all subfields of the splitting field. Here ...
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### Example of a field of characteristic p in which the Frobenius endomorphism fails to be surjective

Let $F$ be a field of characteristic $p$. Then the $p$-th power map, $x\mapsto x^p$ is called the Frobenius endomorphism. If $F$ is a finite field, this is an automorphism. Injectivity of this map is ...
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### Prove that $D$ is a field. [duplicate]

Here is the question I want to answer: Let $F$ be a field and $D$ an integral domain which is a finite dimensional vector space over $F.$ Prove that $D$ is also a field. Here are my thoughts: Since $F$...
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### Galois group of $\mathbb{Q}(\sqrt{a+d\sqrt{b}},\sqrt{a-d\sqrt{b}})$.

I was reading these lecture notes of Miles Reid: https://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf on page 47, he writes example 3.21 of $\mathbb{Q}(\sqrt{a+\sqrt{b}},\sqrt{a-\sqrt{b}})$, but he ...
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### Does the discriminant of a polynomial depend on the underlying field?

Say I want to show the discriminant of $X^4 + rX + s \in \mathbb{Q}[X]$ is $-27r^4 + 256s^3$. I can do this by showing it's a symmetric polynomial in r,s so by total degree in the roots it's a linear ...
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### Resolvents of quartic polynomials

Let $f(x)\in\mathbb{Q}[x]$ be monic irreducible of degree $4$, with $$f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\alpha_i\in\mathbb{C})$$ In ...
Suppose that $K_1$, $K_2$ and $K_3$ are finite Galois extensions of a field $k$. Let $(K_1 \cap K_3)(K_2 \cap K_3)$ be the compositum of $(K_1 \cap K_3)$ and $(K_2 \cap K_3)$. Is it always true that $$... 1 vote 0 answers 31 views ### Radical Solvability by pth roots The characterization of field extension radical solvability in terms of the Galois group is fairly straightforward and well-known. I'm curious about a more specific problem. Namely, given a field F ... 2 votes 1 answer 23 views ### Degree of subfield fixed by single automorphism Let L/K be a finite Galois extension. If \sigma \in \mathrm{Gal}(L/K) has order d, is it the case that$$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\} satisfies $[L^\sigma:K] = d$? This ...
I am studying Galois theory.I am following the lecture series of K.Hanumanthu on Introduction to Galois Theory.He has started the topic with the definition of group characters. Defn. Let $G$ be a ...