# Tagged Questions

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, ...

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### Degree of the extension field

Aren't we supposed to know the degree of the field extension to solve this problem? Did I miss something?
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### Show that a polynomial is still irreducible in a extension field

I have found this question on the Papantonopoulou's Algebra book: Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ with deg$f(x) = 15$ and deg$g(x) = 14$. Let $\alpha$ be a ...
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### Fixed field of an automorphism of $\mathbb{C}$ extended from and automorphism of a subfield.

Let us consider the field isomorphism $$\tau:\mathbb{Q}(\sqrt{2})\rightarrow\mathbb{Q}(\sqrt{2}), a+b\sqrt{2}\mapsto a-b\sqrt{2}.$$ I have read that any isomorphism of a subfield of $\mathbb{C}$ can ...
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### Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
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### For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
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### On the degree of cyclotomic fields extension.

Let $F$ be a field and $n$ be any integer satisfying $(n,\text{char}(F))=1$ when $\text{char}(F)\ne 0$. Let $\xi$ be a primitive $n$-th root of unity. I know that $\text{Gal}(F(\xi)/F)$ is isomorphic ...
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### Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
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### $K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
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### Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
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### Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
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### Transcendental field extensions obtained by taking quotient of $k[X_1, \ldots, X_n]$

Given a field $k$, $n \in \mathbb{N}\setminus \lbrace 0 \rbrace$ and $M$ a maximal ideal of $k[X_1, \ldots, X_n]$, can the field $L = k[X_1, \ldots, X_n]/M$ ever be transcendental over $k$? By this I ...
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### Interpreting $\sqrt{2}$

My apologies if this question is somewhat vague or too broad. Analytically, the fact that $\sqrt{2}$ is no trivial fact. It requires some kind of completion of the rational numbers, e.g. by adjoining ...
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### Equivalences Galois extension

Suppose $K$ is a field of characteristic $\neq 2$ and suppose $c \in K\setminus K^2$ and $F=K(\sqrt{c})$. Suppose $\alpha=a+b\sqrt{c}$ with $a,b\in K$ so that $\alpha \notin K^2$ and ...
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### If Q(a,b) is a field extension, can we always choose an equivalent extension Q(c) such that c=a+b?

If we have two complex numbers $a,b$ that are algebraic over $\mathbb {Q}$, we can make an extension $\mathbb {Q}(a,b)$ that is equal to an extension $\mathbb {Q}(c)$ for some $c\in \mathbb {C}$. ...
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### Prove $R$ is a field. [duplicate]

Suppose that $K$ is an algebraic extension of a field $F$. Suppose that $R$ is a ring with $F\subseteq R\subseteq K$. I'm trying to prove that $R$ is a field. But I have no idea. Anyone can help me?
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### char$(K)=0$ then $K(x^{2}) \cap K(x^{2}-x)=K$ [duplicate]

Let $K$ be a field of characteristic zero and $K(x)$ the field of rational functions with coefficients in $K$. Let $K(u)$ denote the subfield of $K(x)$ generated by $u \in K(x)$ over $K$. My ...
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### Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
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### A normal closure of an arbitrary field extension

Let $L/K$ be an arbitrary algebraic field extension. How is a normal closure of $L$ (the smallest normal extension of $K$ containing $L$) constructed? If $L/K$ is finite, then writing ...
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### Find $[\mathbb{Q(\alpha)}: \mathbb{Q}]$ where $\alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9$

Suppose $\zeta$ is a primitive $11$-th root of unity and $\alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9$ Find $[\mathbb{Q(\alpha)}: \mathbb{Q}]$ Could someone please give me a hint ...
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### Degree of field extensions

I am struggling to solve these two problems in Galois Theory. Could you help me please? Suppose $K=\mathbb{Q}(\sqrt[5]{3}, \sqrt[5]{7})$ Prove that $[K : \mathbb{Q}]=25$ $K$ is a splitting ...
### Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?
Let $F$ be an extension field over $K$ such that $F(x)$ is a finite extension over $K(x)$ ; then is it true that $[F:K]$ is also finite ? ( I know about the converse , that if $F/K$ is a finite ...