# 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. Please use (galois-theory) instead for questions specifically about that topic. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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### For which numbers $l \leq |K|$ field $K$ has subfield, which has $l$ elements? [duplicate]

char$(K) = p$, $|K|= p^n$, $|K^{\times}| = p^n-1.$ We know that $K^{\times}$ is cyclic group. Let $H$ be a cyclic subgroup of $K^{\times}$. Subgroup of cyclic group is also cyclic, so $H$ is cyclic ...
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### A field extension, the larger field is algebraically closed, are there finite subextensions of arbitrary large degree?

Let $L/F$ be a field extension of infinite degree. Assume that $L$ is algebraically closed. I am not assuming that $L$ is an algebraic closure of $F$. Let $d\geq 1$. Must there be an intermediate ...
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### What is the first order logic statement of “this field is of characteristic zero”?

I want to state that a field $F$ is of characteristic zero in logical notation to an audience without referring them to the meaning of the characteristic of a field. My first thought was the ...
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### Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
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### Normal basis of an extension of degree 4 over its prime field.

I want to construct a normal basis of $\mathbb{F}_{p^4}$ over $\mathbb{F}_p$, where $p$ is an odd prime. Is there any particular method to do it?
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### Given a quadratic extension $K/F$, is there a quadratic extension $L/K$ such that $\operatorname{Gal}(L'/F)=D_4$

Let $F$ be a number field (or a function field with constants $\mathbb{F}_q$ where $2 \nmid q$). If $L$ is some extension such that $[L:F]=4$ and $\operatorname{Gal}(L'/F)=D_4$, the dihedral group ...
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### Find roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}/(x^2+x+8)$

I need to find the roots of primitive polynomial $x^2+4x+2$ in $\mathbb{Z}_{11}$ over field $\mathbb{Z}_{11}/(x^2+x+8)$ ($x^2+x+8$ is also primitive). As far as I'm concerned the answer is the ...
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### $\mathbf{Q}[2^{1/3}w]$ = $\mathbf{Q}[2^{1/3},w]$?

Consider $x^3 - 2$ with roots $2^{1/3}$, $2^{1/3}w$, $2^{1/3}w^{2}$ over $\mathbf{Q}$. where $\mathbf{Q}$ is the set of rationals and $w$ is the cube root of unity. Then which of the following are ...
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### Prove $\mathbb Q[\sqrt[3]{2}]$={$a+b\cdot\sqrt[3]{2}$} is not a field [closed]

Suggestion: Prove that you can't write $\mathbb Q[\sqrt[3]{2}]$ like $a+b\cdot\sqrt[3]{2}$
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### Transcendental Extension is Separable over Separable Closure

Let $K/k$ be a finitely generated field extension with transcendence basis $\alpha_1, \cdots \alpha_r$. I understand that this means $K$ is algebraic over $k':= k(\alpha_1, \cdots, \alpha_r)$ and $K$ ...
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### Separable polynomials by Bourbaki

In Bourbaki, Algebra, V, 37, we have Proposition 3, which states that: Let $f$ be a non-zero polynomial in $K[X]$ and let $\Omega$ be an algebraically closed extension of $K$. The following ...
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### Example of exotic $S_5$ as a Galois group

Is there an example of a sextic irreducible polynomial over $\mathbb{Q}$ with Galois group isomorphic to $S_5$? The transitive action of the Galois group of this polynomial on the 6 roots of $p(x)$ ...
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### $p$'th roots in an extension of prime degree $p$.

Let $F$ be a field of characteristic $0$, and let $p$ be a prime number. Choose an element $r\in F$ and suppose that $r$ has no $p$'th root in $F$. Let $F':=F(\sqrt[p]{r})$, this is an extension of $F$...
### About the reduction modulo $p$ for calculating Galois groups
I know the following theorem: Let $f(x)\in\mathbb{Z}[x]$ be a monic separable polynomial of degree $n$. If $f(x)\mod p$ is separable and irreductible for some prime $p$, then there is a cycle of ...