# 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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### Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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### Elements in a Field of size $27$

I constructed the Field $$F_3[x]/<1 + 2x + x^3>$$ as the question asked to construct a field of size $27$ and I understood everything up to this point. The solution then says the elements in ...
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### Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[...
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### Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, \sqrt{f}]$....
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### About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n$ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n$ ( A subgroup $G$ ...
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### rational numbers field axioms

Let $\mathbb Q$ be the rational number field. Is the group $K=\left\{\left.\begin{pmatrix} a & 2b\\ b & a \end{pmatrix}~\right|~ a,b\in \mathbb Q\right\}$ a field with the regular addition ...
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### Some natural question on subfield of Galois extension

Let $\alpha,\beta\in \mathbb{\overline{Q}}$ and assume $\deg(\text{Irr}(\alpha,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q(\alpha)})=2$. Then I strongly guess that ...
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### If $f(a)=f(a+1)$, then $F$ has characteristic $0$.

Suppose $f\in F[x]$ is irreducible, $E$ is the splitting field of $f$, and for some $a\in E$ we have $f(a)=f(a+1)=0$. Then $F$ has characteristic $0$. I'm not sure how to use the last assumption: If ...
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### Finding the fixed subfield corresponding to a cyclic subgroup of the Galois group

Let's say I have a field extension $E$ of some field $F$ and I also know the Galois group of $E$ over $F$. Suppose I have a subset of this Galois group which is cyclic, thus generated by some ...
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### Intersection of two subfields of $F(X)$ [duplicate]

Let $E=F(x)$ for a field $F$ of characteristic $0$. Show that $F(x^2) \cap F(x^2-x) = F$ as subfields of $F(x)$. I could use a hand with this... Thanks
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### Show that $x^{n-1} =1$ for all non-zero element in a field

Question: Show that $x^{n-1} =1$ for all non-zero element in a field Let F be a finite field of order n. Show that $x^{n-1}=1$ for all non-zero $x \in F$. We have $\left | F \right |=n.$ ...
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### Give all extensions of the mapping to an isomorphic mapping

Let $E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}).$ It can be shown that $[E : \mathbb{Q}] = 8.$ For each given isomorphic mapping of a subfield of $E,$ give all extensions of the mapping to an ...
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### Prove that $\sigma(x)$ and $x$ are conjugate over $F$

Let $E$ be an algebraic extension $F$ and $x \in E$ and $\sigma: E \to E$ be an automorphism of $E$ fixing $F.$ Prove that $\sigma(x)$ and $x$ are conjugate over $F.$ I am starting ...
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### Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
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### Why is the Galois group for a cyclotomic extension isomorphic to $Z_n^{\times}$ and not to $Z_{n-1}$?

I'm struggling to understand this. Let $\mathbb{Q}(\xi)$ be the splitting field for $x^n - 1$, so here $\xi$ is the primitive n-th root of unity. If I consider the possible automorphisms of this ...
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### Is $\prod_{\sigma\in \Sigma(L)}\mathbb Z$ free of rank $[K:\mathbb Q]$?

Let $L/\mathbb K$ be a Galois extension of number fields and $\Sigma(L)$ the set of embeddings of $L$ into $\mathbb C$. Let $H_L= \prod_{\sigma\in \Sigma(L)}\mathbb Z$. Let $G$ act on $H_L$ in the ...
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### Proof of $\Bbb R$ is the unique complete linear order.

I'm looking for the theorem that says that all linearly ordered, complete fields are isomorphic. I couldn't find references online, but I'm sure this theorem must have some name. A link would be ...
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