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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The fixed field of $A$ is equal to the fixed field of $\langle A\rangle$.

Let $E$ be a finite extension field of $F$. Let $A$ be a subset of $Gal(E/F)$. Let $\langle A\rangle$ be the subgroup generated by $A$. Is the fixed field of $A$ equal to the fixed field of $\langle ...
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0answers
18 views

How are subgroups and subfields related in Galois theory

My teacher for Galois theory said it was important not to mix up groups and fields within Galois theory Can someone explain how they are related and how to distinguish the Galois group and Galois ...
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0answers
17 views

If K is a field whose characteristic is not 2, show F/K is Galois [duplicate]

Let F/K be a field of extension 2, If K is a field whose characteristic is not 2, show F/K is Galois. I think I need to use a fact that the extension F/K is Galois if and only if K is the splitting ...
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21 views

Complex extension isomorphic to $\mathbb{R}$?

Let $K$ be some field extension of $\mathbb{Q}$ containing some complex number $c=a+bi$ with $a,b\in \mathbb{R}$ and $b\neq 0$. Is it possible that $K\cong \mathbb{R}$ as fields? I tried to disprove ...
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1answer
25 views

Can there be a numerical system in which logarithms can be expressed in terms of exponents in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
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0answers
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perfect squares in $\mathbb{Q}(\sqrt{2},\sqrt{91})$

Given a polynomial $P(X)$ with coefficients in $\mathbb{Q}(\sqrt{2},\sqrt{91})$ how do I find values of $X$ in $\mathbb{Q}(\sqrt{2},\sqrt{91})$ such that $P(X)$ is a perfect square in ...
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Computing the degree of the splitting field of $x^3+18x+3$ over $\Bbb Q$

Let $T$ be a splitting field of polynomial $$f(x)=x^3+18x+3\in\mathbb{Q}[x].$$ What is the degree $[T:\mathbb{Q}]$? My thoughts: the polynomial $f$ is irreducible over $\mathbb{Q}$, therefore the ...
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1answer
42 views

Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
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22 views

All possible degree 3 field extensions

Do we know anything about all possible degree 3 field extensions? If characteristic is 3: If Galois, then Artin-Schreier. If inseparable, then this is just cube root. If separable, but not normal, ...
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1answer
28 views

Prove that $e^{2\pi i/5}$ is not in the $7$-th cyclotomic field.

Let $\xi_n = e^\frac{2\pi i}{5} $. Prove that $\xi_5 \notin \Bbb{Q}(\xi_7)$ where $\Bbb{Q}(\xi_7)$ is the 7-th cyclotomic field. How would I approach this question? I'm having a difficult time coming ...
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If $E$ is finite, show that $E=F(u)$ for some $u\in E$.

Let $E\supseteq F$ be fields. If $E$ is finite, show that $E=F(u)$ for some $u\in E$. What I've come up with so far: Let $u\in E$ be the primitive element of $E$. Then $E^*=\langle u \rangle$. So ...
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1answer
20 views

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$.

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its decomposition field. Show that $[F:K]$ divides $n!$. Here $[F:K]$ denotes the dimention of $F$ over $K$ as a ...
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3answers
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Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $E \subset \mathbb{R}$ a field containing $\mathbb{Q}$ such that the extension $E \vert \mathbb{Q}$ is Galois, and let $F := E(\sqrt{-1})$. Show that the extension $F \vert \mathbb{Q}$ is ...
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0answers
13 views

degree of a field extension and generator

I assume that $\alpha$ is algebraic over F and then $F(\alpha)$ is just a simple extension. And then I am able to find a basis for both $F(\alpha)$ and $F(\alpha^3)$. Then we can apply the theorem ...
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1answer
69 views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] ...
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1answer
30 views

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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1answer
37 views

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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0answers
31 views

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 ...
2
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0answers
70 views

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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2answers
26 views

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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1answer
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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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1answer
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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 ...
4
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1answer
60 views

$K/\mathbb{Q}$ 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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1answer
38 views

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 ...
2
votes
1answer
66 views

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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0answers
13 views

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 ...
2
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1answer
23 views

Which cyclotomic fields are different?

For $n$ a positive integer, let us write $\zeta_n = e^\frac{2 \pi i}{n}$, a primitive $n$th root of unity. It is clear that, if $m$ divides $n$, then we have an inclusion of cyclotomic fields $$ ...
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1answer
14 views

Systematic way of expressing field extensions

If a field $Q$ were to be extended to include roots of the quadratic polynomial $x^2$$-2=0$, the extended field $Q$($\sqrt2$) would include elements of the form $a$ + $b$$\sqrt 2$. However, extending ...
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1answer
28 views

Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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Separable and purely inseparable extensions over fields of characteristic $p$

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $E/F$ a finite extension. For $n \in \Bbb Z_{\geq 0}$, set $E^{p^{n}}=\{ x^{p^{n}} \mid x \in E \} \subset E$. My question is the ...
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3answers
24 views

Ring Extension: Mapping: $ \mathbb Q[\sqrt d] \rightarrow \mathbb Q$

Show that the Norm: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q, (r+s\sqrt d) (r-s\sqrt d) = r^2 - ds^2$ is multiplicative, i.d. $N(xy) = N(x)N(y)$ How to show it without computing? (I tried to do it ...
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0answers
40 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
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0answers
19 views

When are composite extensions isomorphic?

Let $E$ and $F$ be two totally complex finite extensions of $\mathbb{Q}$, let $\sigma_i \, :\,E \rightarrow \mathbb{C}, i\in I$ and $\tau_j \,: \,F \rightarrow \mathbb{C}, j\in J$ denote all their ...
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2answers
6 views

degree of [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]

I got maybe easy problem. I am not sure if it is true that [$\mathbb Z_2[x]/f\mathbb Z_2[x]: \mathbb Z_2$]=deg $f$ where $f \in \mathbb Z_2[x]$ irreducible. Can anybody help me ? Thanks
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1answer
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Extension Field of $e^{\frac{i\pi}{5}}$

Let $w = \cos\frac{2\pi}{10}+ i \sin\frac{2\pi}{10}$. Let $K=\mathbb Q(w^2)$ and $L=\mathbb Q(w)$. Then $[L:\mathbb Q] = 10$ $[L:K] = 2$ $[K:\mathbb Q] = 4$ $L = K$ Now, once we know the ...
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1answer
23 views

Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
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1answer
21 views

$B/K$ is a field extension of degree 1 implies $B = K$?

I was reading the fundamental theorem of Galois theory. Here's an excerpt. Theorem. Let $E/F$ be a finite Galois extension, then $$ \varphi: K \mapsto Aut(E/K) $$ and $$ \psi: H \mapsto ...
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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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1answer
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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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2answers
42 views

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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1answer
36 views

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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0answers
28 views

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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2answers
59 views

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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0answers
14 views

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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2answers
71 views

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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30 views

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 ...
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1answer
21 views

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 ...
0
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1answer
16 views

Proving an element belongs to field extension

I am unsure of questions asking to prove that an element belongs to a field extension. Here is an example: Prove that $\sqrt2 \in \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$ $\sqrt2 \notin ...
3
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1answer
31 views

If $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ and $[K : \mathbb{Q}]=p$ then $K= \mathbb{Q}(\sqrt[p]{2})$

Let $p,q$ be distinct prime numbers and assume that $p<q$. Prove that if a subfield $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ satisfies $[K : \mathbb{Q}]=p$, then $K= ...