# Tagged Questions

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....

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### The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
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### Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem. If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
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### Is there a purely algebraic proof of the Fundamental Theorem of Algebra?

Among the many techniques available at our disposal to prove FTA, is there any purely algebraic proof of the theorem? That seems reasonably unexpected, because somehow or the other we are depending ...
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### Field with natural numbers

To make sure that we are talking about the same, I would like to post the relevant definitions I know first. Definitions: A pair $(G, +)$ where $G$ is a set and $+: G \times G \rightarrow G$ is ...
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### Constructing a Galois extension field with Galois group $S_n$

Constructing a Galois extension field $E$ with $Gal(E/F)= S_n$ How do I construct one?
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### How to prove that the Frobenius homomorphism is surjective?

$R$ is a domain with characteristic $p$ ($p$ is prime).There is a homomorphism $f : R \to R$, $f(a)=a^p$. $f$ is called the Frobenius homomorphism. And I have known this. When $R$ which is mentioned ...
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### How to show that a finite commutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
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I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 \... 1answer 177 views ### Intermediate fields between$\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$and$\mathbb{Z}_2 (x,y)$Let$K=\mathbb{Z}_2 (x,y)$, where$x,y$are independent, and$L$be a splitting field extension of$(X^2 - x) (X^2 - y)$, then$[L:K] = 4$and$L = K(\sqrt{x},\sqrt{y})$where$\sqrt{x},\sqrt{y}$are ... 3answers 583 views ### On the meaning of being algebraically closed The definition of algebraic number is that$\alpha$is an algebraic number if there is a nonzero polynomial$p(x)$in$\mathbb{Q}[x]$such that$p(\alpha)=0$. By algebraic closure, every nonconstant ... 1answer 3k views ### Existence of irreducible polynomials over finite field Let$F$be a finite field. How do we prove that for each$n \in \mathbb{N}$there is an irreducible polynomial of degree$n$? One can assume that$F = \mathbb{F}_{p^m}$where$p$is prime. If$n \ge |...
I am trying to prove the following: Let $K=\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n})$. Show that $K/\mathbb{Q}$ is Galois with Galois group $(\mathbb{Z}/2\mathbb{Z})^n$. I have attached my proof ...