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 ...

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$\mathbb Q[e^{\frac{2\pi i}{3}}] \cong\mathbb Q[x]/(x^2+x+1)$

I want to show that the field $\mathbb Q[e^{\frac{2\pi i}{3}}]$ is isomorphic to $\mathbb Q[x]/(x^2+x+1)$. Can someone give me a hint to approach this?
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Unique degree $d$ monic polynomial $f$ with $\mathbb{Q}$-coefficients such that $f(\beta) = 0$, $f$ irreducible. [closed]

If $\beta \in \mathbb{C}$ is algebraic (over $\mathbb{Q}$) and $\mathbb{Q}[\beta]$ has $\mathbb{Q}$-dimension $d$, prove that there exists a unique degree $d$ monic polynomial $f$ with ...
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Proving the uniqueness of the additive inverse in a field without the commutative property

Someone told me that it was possible to prove that 0 is unique without using the commutative property. I don't see how, and I constructed a multiplication/addition table with elements $0_1, 0_2$, and ...
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$F$ a field over $\mathbb{Q}$ of dimension $2$, show $a \in F$ satisfies $a^2 - n = 0$

Let $F$ be a field with $\Bbb{Q} \subseteq$ $F$. If $F$ considered as a vector space over $\Bbb{Q}$ has dimension 2, show that there exists an element $a \in F$ which is not in $\Bbb{Q}$ which ...
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commutative ring with id without non-trivial ideals is a field. Why?

Firstly, I dont have any intuition to this exercize. I mean let look at R. It is a field, despite the fact that there are a lot of nn-trivial ideals. So from first look, I dont see reason, why ...
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A different viewpoint of Hilbert's Theorem 90

Let $L/K$ be a galois extension with galois group $G$($|G| = n$) cyclic and generated by $\sigma$. Let $\beta \in L$ have $N(\beta) = 1$. $N(.)$ is the norm function from $L$ to $K$. Hilbert's ...
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How to find all intermediate fields of $\mathbb{Q}(2^{1/4})$ over $\mathbb{Q}$ without the Galois correspondence?

As a $\mathbb{Q}$-vector space, $\mathbb{Q}(2^{1/4})$ has a dimension 4. So, any intermediate subfields have a dimension 2 over $\mathbb{Q}$. I'm already know that $\mathbb{Q}(2^{1/2})$ is such a ...
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What are all the automorphisms of $\mathbb{Q}(\sqrt{2})$? [closed]

The field $\mathbb{Q}(\sqrt{2})$? is defined as $\{a+b\sqrt{2}: a, b \in \mathbb{Q}\}$. Are there only two automorphisms, one mapping to $\{a+b\sqrt{2}\}$ and the other mapping to $\{a-b\sqrt{2}\}$? ...
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Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$

This was an assertion made in our textbook but I have no idea how to show that either statement is true. Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, ...
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$\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$

Prove that $\mathbb{Q}\left(\sqrt{p}\right)=\{a+b\sqrt{p}\mid a, b \in\mathbb{Q}\}$ is a subfield of the field $\mathbb{R}$, where $p$ is a prime number I know this is true for many primes that ...
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Finitely generated modules over principal ideal domain

Let $A$ be principal ideal domain with field of fractions $K$. $L$ is finite separable extension of $K$ and $B$ is the integral closure of $A$ in $L$. It is obvious that there exists a constant $d$ in ...
So the question is: Let $\sigma$: $F_1 \xrightarrow[]{} F_2$ be a homomorphism where $F_1$ and $F_2$ are fields. Show $\sigma$ induces an isomorphism between their prime subfields and, in ...