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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$\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
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What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?

I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$? Here's what I've got so far: ...
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Why cyclic subgroup of $\text{Gal}(L/K)$ is achieved by some decomposition group?

Let $L/K$ be a finite Galois extension. Let $\text{Gal}(L/K)$ be its Galois group. For every cyclic subgroup $H$ of $\text{Gal}(L/K)$, why does there exists a prime $v$ of ring of integers of $K$ and ...
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Concerning some quadratic field extension $[L:\mathbb{C}(u,v)]=2$

In the following question I am actually asking about the answer to this MO question. First, I will ask it generally, then I will present the question and answer appearing in MO. A general question: ...
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Conway Polynomial for p=2, n=3?

Im doing an exercise on Conway polynomials. As far as im concerned, for p=2, n=3 both $f(x)=x^3 + x^2 + 1$ and $g(x)=x^3 + x + 1$ satisfy every condition. According to every source i found, the latter ...
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How to show that the field $\mathbb{F}_2(t^{\frac{1}{3}})$ does not contain a third root of unity

I want to find a non-normal extension in characteristic p. The following extension is not normal : $$\mathbb{F}_2(t)\subset \mathbb{F}_2(t^{\frac{1}{3}})$$ It's minimal polynomial is : $X^3-t$ and it'...
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Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
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Let $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle$. How many elements of order 2 and 3 are there in the additive group of $E$? How many generators?

I know that for $E=\mathbb{Z}_3[x]/\langle x^2+x+2\rangle$, since $x^2+x+2$ is irreducible over $\mathbb{Z}_3$, $E$ is a field, because $\langle x^2+x+2\rangle$ is a maximal ideal. In addition, $E$ ...
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An exercise about abelian Kummer extensions

I'm trying to do this problem about abelian Kummer extensions: Image transcript and my attempts are below: Let $K/F$ be a Galois extension with Galois group $G=\operatorname{Gal}(K/F)$ of order $n$. ...
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Understanding a proof in JS Milne's Fields and Galois Theory (Prop 7.10)

The following is proposition 7.10 in Milne's Fields and Galois Theory: Let $G$ be a group of automorphisms of a field $E$, and let $F=E^G$ (ie. $F$ is fixed field of $G$). If $G$ is compact and the ...
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An algebra structure over a perfect field

When I read some paper, I see the following word: When $k$ is a perfecr field, a finite dimension $k$-algebra $A$ can be writted as $S\oplus \rm{rad}(A)$, where $S$ is the maximal semisimple ...
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Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
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Primitive elements of the rationals function extension [duplicate]

I want to prove this: Let $k$ a field, $E:=k(x)$ the quotient field of $k[x]$ and $u\in E$. Show tha $E=k(u)$ if, and only if, $u=\frac{ax+d}{cx+d}$ for some $a,b,c,d\in k$ such that $ad-bc\neq 0$. I ...
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"field" vs. "vector field" [duplicate]

Is the "field" in the "vector field" as the same "field" in algebra: as the commutative ring with the multiplicative inverse? If yes, then the "vector field" ...
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Irreducibility of a Polynomial with Prime Exponents

Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$ where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
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Does $Z(f) \cap Z(g) = \emptyset$ implies $\gcd(f,g) = 1$? [duplicate]

If $f(x)$ and $g(x)$ are in $\mathbb{F}[x]$, where $\mathbb{F}$ is a field. Let us assume that $Z(f) = \{x \in \overline {\mathbb{F}} : f(x) = 0\}$, where $\overline{\mathbb{F}}$ is the algebraic ...
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How to prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$ using elementary techniques?

Prove $\sqrt{2} \notin \mathbb{Q}(\sqrt[3]{2})$. My try by contradiction: Assume there exist $a,b,c \in \mathbb{Q}$ such that: $\sqrt{2}=a + b \sqrt[3]{2} +c \sqrt[3]{4}$. Squaring both sides, we ...
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Question in an example of prime decomposition in an extension of DVRs.

Choose any element $\alpha \in \mathbb{Q}[[x]]$ such that it is transcendental over $\mathbb{Q}(x)$(This is possible by comparing cardinalities.), and let $\beta=\alpha^2$. Let $K=\mathbb{Q}(x,\beta)$,...
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How to prove that two algebraically closed field with same characteristic and cardinal are isomorphic? [duplicate]

Hi I would like to prove that for algebraic closed field $F_1,F_2$ if: They have the same characteristic $p$ They have the same cardinal and is uncountable Then they are isomorphic. I have proved ...
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Order of $\mathbb F _p [x] / (f)$.
I could use some help with the following exercise: Find the number of reducible monic polynomials of degree $2$ over $\mathbb F_p$. Show this implies that for every prime $p$ there exists a field of ...