# Questions tagged [extension-field]

Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.

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### An extension corresponding to a subgroup of Galois group

Let $G$ be the Galois group of $f(x)=x^6-2x^4+2x^2-2$ over $\mathbb{Q}$. Describe an extension corresponding to any of it's proper subgroups of maximal order (i.e. find generators of this extension). ...
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### How to find the degree of the extension $[\mathbb{Q}(\sqrt{3+2\sqrt{5}}):\mathbb{Q}]$?

How to find the degree of extension for $[\mathbb{Q}(\sqrt{3+2\sqrt{5}}):\mathbb{Q}]$? I believe that the minimal polynomial of $\sqrt{3+2\sqrt{5}}$ is $x^8-6x^4-11$, but I don't know how to ...
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### $Q\subset L$ with $G := \text{Gal}(L/Q)$, Is $L$ contained in the field of constructible numbers? [closed]

$Q \subset L$ is a finite Galois extension with $G := \text{Gal}(L/Q)$ and $G$ is isomorphic to $S_3$, the symmetric group on $3$ elements. Is $L$ contained in the field of constructible numbers?
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### Representatives of residue classes as elements of a vector space

Let $K$ be a number field of degree $n$. Let also $\mathcal{O}(K)$ be the ring of integers of $K$ and let $B$ be a basis for $\mathcal{O}(K)$. Given $q\in K$ we write $\mathbf{q}$ for the column ...
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### Scalar restriction of bilinear maps

Let $R$ be a ring (commutative and with unity), $S\subset R$ be a subring. Consider three $R$-modules $M$, $N$ and $Z$. Let $\operatorname{Hom}_R(M\otimes_RN;Z)$ be the R-module of $R$-bilinear maps ...
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### Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$

Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ ...
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### Degree of an exclusionary Field Extension

Let's say I've got a field $\mathbb{Q}[i]$\ $\mathbb{Q}$. What's the degree of the field extension $\mathbb{Q}[i]$\ $\mathbb{Q}$ : $\mathbb{Q}$? Clearly without the exclusion this has a degree of 2; ...
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### Prime decomposition of pR in $\mathbb{A}\cap \mathbb{Q}[\alpha]$ for $\alpha={^3\sqrt{hk^2}}$ if p is a prime such that $p^2|m$

I'm going through Marcus number Field chapter 3 an I'm finding very hard to understand the part about the decomposition of pR (theorem 27) that tells us that if $p\not||R/Z[\alpha ]|$ then we can ...
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### Is every field a field extension of some form.

I am new to finite field theory .While I was going through the theory I figured that $\mathbb{C}$ is in fact $\mathbb{R}(i)$ isomorphic to $\mathbb{R}[x]/(x^2+1)$ .So I had a question in mind is ...
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### Galois Group of $x^{6}-2x^{3}-1$

I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt{1+\sqrt{2}}$. I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
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### How to show $E\otimes_k\bar k$ has at least two prime ideals? [closed]

Suppose $k\subsetneqq E$ are two fields, and $E$ is separable over $k$, $\bar k$ is the algebraic closure of $k$, how to show $E\otimes_k\bar k$ has at least two prime ideals?
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### Prove that $\mathbb{Q}[\sqrt{p}, \sqrt{p^2}]$ is a field.

$$\mathbb{Q}[\sqrt{p}, \sqrt{p^2}] := \{a+b\sqrt{p}+c\sqrt{p^2}: a,b,c \in \mathbb{Q}\}$$ That's the description of the wanted set of numbers. a, b, c are its elements. If the note isn't ...
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### Calculate the degree of a composite field extension

I am working on this problem : Let $a>1$ be a square-free integer. For any prime number $p>1$, denote by $E_p$ the splitting field of $X^p-a \in Q[X]$ and for any integer $m>1$, let $E_m$ ...
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### If $K=\{a+b\sqrt2\mid a,b\in\mathbb{Q}\}$, find $[K ∶\mathbb{Q}]$ and $[K(\sqrt3) ∶\mathbb{Q}]$. [closed]

I'm new to the subject and struggling to understand the steps when finding the degree of a field extension, I've been finding the minimal polynomial and then using the degree of that as the answer, ...
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### Field extension of degree three

It's well known that $$\mathbb{Q}(\sqrt{2},\sqrt{5}) = \mathbb{Q}(\sqrt{2} + \sqrt{5})$$ This property is also true with the cubic root (for a great general theorem), but I want to prove this via an ...
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### How are these two lemmas about simple algebraic extensions and polynomials restatements of each other?

Lemma 5.14. says Let $K(\alpha) : K$ be a simple algebraic extension, let the minimal polynomial of $\alpha$ over $K$ be $m$, and let $\partial m = n$. Then $\{1,\alpha,\ldots,\alpha^{n-1}\}$ is ...
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### The number of intermediate field extensions $\mathbb{Q}(ζ_{630})/M/\mathbb{Q}$ of $[M:Q]=3$

I'm trying to solve this exercise: Find the number of intermediate field extensions $\mathbb{Q}(ζ_{630})/M/\mathbb{Q}$ satisfying $[M:Q]=3$. Since \begin{gather} Gal(\mathbb{Q}(ζ_{630})/\mathbb{Q})...
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### Degree of separability equals degree of separable closure

For a field $K$ of characteristic $p > 0$, and a finite field extension $L/K$, let $K_s$ be the separable closure of $K$ in $L$. I am to show that $[L : K]_s = [K_s : K]$. This would mean ...
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### $E_1/F$ and $E_2/F$ finite field extensions, does degree of compositum $E_1E_2$ over $F$ divide the product $[E_1:F] [E_2:F]$?

Suppose $E_1/F$ and $E_2/F$ are finite field extensions. The degree of the composite field $E_1E_2$ over $F$ is less or equal to the product of the degree of $E_1$ over $F$ times the degree of $E_2$ ...