# 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.

720 questions
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### The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ ...
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### What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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### Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$. What is the intersection $F_\infty\cap K_\infty$? (Here $\zeta_{2^n}$ is a ...
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### On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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### Example of non-simple extension $K/F$ such that the purely inseparable closure of $F$ in $K$ is simple

The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely ...
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### Splitting field over field extenstion

this is my first question here, I'll apologise in advance for any kind of noob mistakes because I'm aware that I might to do them. I'm solving one assignment for my studies and I can't do anything ...
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### Show that the $\mathbb{Q}$-homomorphism is well-defined.

Suppose that $a \in \mathbb{C}$ is algebraic over $\mathbb{Q}$ with $p(x) = \min(\mathbb{Q},a)$, and let $b$ be any root in $\mathbb{C}$ of $p$. Show that the map $\sigma:\mathbb{Q}(a) \to \mathbb{C}$ ...
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### Galois groups of quartic over $\mathbb Q$ and $\mathbb Q(i)$

Let $f(x) = x^4 - 4x + 2$. I am asked to find its Galois groups over $\mathbb Q$ and $\mathbb Q(i)$. I believe I have done this, but have arrived at a result that I was not expecting, so I wanted ...
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### Find Galois group and all intermediate fields of the extension $L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$

Find Galois group, all subgroups and the corresponding subfields of the following extension : $$L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$$ 1) I found the degree of the extension $$[L:K]=8$$ ...
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### Proving that $\sum_{v\in M_K, \,\,v\text{ archim.}} n_v=[K:\mathbb{Q}]$

Let $L/K$ be an extension of number fields, $M_K, M_L$ complete sets of representatives of places at $K$ and $L$, respectively. I'm familiar with the formula: \sum_{w\in M_L, \,\,w|v} n_w=[L:K]n_v\,...
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### Determining subfields of cyclotomic field extension

I am trying to determine the number of subfields of $\mathbb Q (\zeta_{16})$ for which $[M :\mathbb Q]=2$ and their structure. (Where $M$ is a subfield and $\zeta_n$ is $n$th primitive root of unity ) ...
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I am trying to understand a proof of the following theorem: Theorem: Let $\alpha$ be an algebraic integer wich has degree $n$ over $\mathbb{Q}$. Then the field extension $\mathbb{Q}\left (\alpha\... 0answers 67 views ### Degree of splitting field extension of a polynomial over$K=\mathbb{F}(x)$Let x be transcendent over$\mathbb{F}$, where$\mathbb{F}$is a field with characteristic 2. I have a polynomial$f=t^3+xt+x\in K[t]:=\mathbb{F(x)}[t]$. Let L be the splitting field of$f$. I know ... 0answers 61 views ### Galois theorem in general algebraic extensions I have proved for myself the following theorem, generalizing Galois theorem to general algebraic extensions. My question is: is it true, and is there some reference to this theorem in the literature? ... 0answers 302 views ### (Proof Verification) Prove that the sum of two algebraic numbers is algebraic. Here is my proof for the statement that the sum of two algebraic numbers is algebraic: Let$\alpha,\beta$be algebraic numbers. Then we know that$[\Bbb Q(\alpha):\Bbb Q]$is finite. Similarly,$[\...
The standard method (I think) for proving that decomposition fields of polynomials exist is by considering an irreducible polynomial $P$ (over the field $K$), and then noticing that $X + (P)$ is a ...
EDIT-1: Since I come up with a proof, I change this previously unanswered question into a proof-verification question. Let $E / F$ be finite field extension. Define \$G(E / F) = \{\phi: E \xrightarrow{...