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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Computing a Galois group

Let $K=\mathbb Q$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. I want to calculate the Galois group $\text{Gal}(L/K)$. The zeros of $f$ are obviously ...
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1answer
13 views

Proposition about of Purely inseparable extension

In the Book's Algebra, Lang, page 251. Proposition: Let $E^p$ denote the field of all elementos $x^p$, $x\in E$- Let $E$ be a finite extension of $k$. If $E^pk=E$, then $E$ is separable over $k$. ...
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1answer
58 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if ...
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1answer
53 views

Galois theory with the extension $\mathbb Q(\alpha)/\mathbb Q$

Let $\mathbb Q(\alpha)/\mathbb Q$ be an algebraic exntension with $\alpha=\sqrt{2+\sqrt{2}}$. 1) Show that the extension is a galois extension (normal and separable) 2) Show that $Gal(\mathbb ...
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1answer
56 views

Roots of unity in splitting field of all polynomials of given degree

Let $K$ be the splitting field of all polynomials of degree $4$ in $\mathbb{Q}[x]$. For which $n\in \mathbb{N}$ the $n$-th primitive root of unity $\xi_n\in K$ ? I've shown that $K$ is an algebraic, ...
2
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3answers
52 views

Writing elements of field extension in terms of the basis determined by a root of a polynomial

Let $\alpha \in \mathbb{C}$ be a root of the irreducible polynomial $$f(X) = X^3 + X + 3$$ Write the elements of $\mathbb {Q}(\alpha)$ in terms of the basis $\{1, \alpha, \alpha^2\}$. The first ...
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1answer
38 views

proof factoring of $x^n-1$

I have a question about the proof of the theorem stating that $$x^n-1 = \prod_{i \in I} m^{(i)}(x)$$ is a factorisation in irreducible factors of $x^n-1$ over a field $\mathbb{F}_q$. Here $m^{(i)}$ is ...
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1answer
61 views

The Kahler differential is zero

While studying for my commutative algebra exam, I have come across this problem. Let $K$ be a field. Let $A$ be a $K-$algebra, finitely generated as a $K$-vector space. Prove that $\Omega^1_{A/K} ...
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1answer
46 views

homomorphisms of field extension

Let $\bar{\mathbb Q}$ be an algebraic closure of $\mathbb Q$. Determine all homomorphisms from $\mathbb Q(\sqrt[4]{2},i)\rightarrow\bar{\mathbb Q}$ and their images! Now the minimal polynomials for ...
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Splitting field and Galois group of a quintic polynomial over $\mathbb{Q}$

On an old algebra prelim, there is a particular problem I would like some help on. It is a five-part question. Let $K$ be the splitting field over $\mathbb{Q}$ of the polynomial $$f(x)=x^5- x^4 + x^3 ...
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0answers
26 views

Mistake in proof that a polynomial $f$ irreducible in $F$ is irreducible in $E$ if $\gcd(\deg f, [E:F])=1$

This is a problem in James Milne's text on Galois Theory: Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F] = m$. If $\gcd(m,n)= ...
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2answers
50 views

Galois-theory - a question about the galois group

I am dealing with galois theory at the moment and I came across with an example in the lecture and I got a question: Let $K=\mathbb Q$ and $L=\mathbb Q(\sqrt{2},\sqrt{3})\subset \mathbb C$. Lets ...
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0answers
21 views

Extension Fields, complex numbers.

I have a question about the complex numbers given as an extension field. I know that complex numbers can be seen as $\mathbb{R}\left[x\right]/<x^2+1>$ (Well, I really don't know it yet, because ...
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1answer
62 views

Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/4\mathbb{Z}$ that contains $\mathbb{Q}(\sqrt{3})$?

Does there exist a normal extension $L ⊃ \mathbb{Q}(\sqrt3) ⊃ \mathbb{Q}$ with Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$?
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0answers
30 views

Show that $K(\alpha,\beta)/K$ is simple

Let $L=K(\alpha,\beta)$ be an algebraic field extension, with $\alpha$ separable over K. Show that $L/K$ is simple. My attempt: If we could show that $L/K$ is finite and separable then the claim ...
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1answer
29 views

fraction field of polynomial ring that is a finite extension of the base field

Let $k$ be a field. Let $P$ be a prime ideal of $k[x_1, ..., x_n]$. Let $K$ be a field of fractions of $k[x_1, ..., x_n]/P$. Suppose $K$ is a finite extension of $k$. Does it then follow that $P$ is ...
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1answer
50 views
+50

The equivalent definitions of linearly disjoint field extensions

I am trying to proove that the following charactrerization of linearly disjontness holds. We have the definition: Given the following extensions, $K\subset L, M \subset N$, $M$ is said to be ...
2
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0answers
16 views

Connection between field and the frobenius homomorphism [duplicate]

Let K be a field with char(K)>0. How do I prove that every algebraic extension of K is a separable extension if and only if $\phi:x \rightarrow x^p$ is surjective ?
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5answers
200 views

Is $\sqrt[3]{5}$ in $\mathbb Q(\sqrt[3]{2})$?

I am trying to solve question 3.7 (b) from Chapter 15 in Artin's book "Algebra". The problem is: Is it true that $\sqrt[3]{5}\in \mathbb Q(\sqrt[3]{2})$? It is clear by Eisenstein's criterion ...
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1answer
45 views

An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
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2answers
38 views

Fixed field of a subgroup of a Galois group

For the Galois group $Gal(\mathbb{Q}(\sqrt2, \sqrt3, \sqrt5)/\mathbb{Q})$, I'm trying to understand how to find the permutations of the roots and how the subgroups of the Galois group are related ...
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1answer
20 views

Problem involving a tower of fields with an algebraic and a normal extension

I seem to be stuck on the following problem about field extensions from an old prelim exam in algebra. Let $K$ be an algebraic field extension of a field $F$ and let $L$ be a subfield of $K$ such that ...
2
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1answer
30 views

Field extensions and minimal polynomials

Let $L/K$ be a field extension and $\alpha,\beta \in L$. Let $f\in K[X]$ be the minimal polynomial of $\alpha$ and $g\in K[X]$ be the minimal polynomial of $\beta$ Show the following: $$f \text{ is ...
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1answer
21 views

Is $\mathbb{R}$ a maximum Dedekind complete field?

Let $\mathbb{R}\subseteq F$ be an ordered Dedekind complete Field (every Dedekind cut of $F$ is already in $F$), does this mean $\mathbb{R}=F$?
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Transcendental extension and algebraic closure

Let $t$ be a transcendental element over field $K$, and let $F/K(t)$ be finite extension. Assume that $E$ is algebraic closure of $K$ in $F$. How to prove that $E/K$ is finite and $[E:K] \mid ...
2
votes
3answers
103 views

Field extensions with(out) a common extension

Let $K$ be a field having two field extensions $L\supseteq K$ and $M\supseteq K$. Does there exist a field $N$ along with embeddings $L\to N$ and $M\to N$, such that the diagram $$ \require{AMScd} ...
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0answers
58 views

Why $\sqrt{5}$ doesn't lie in $\mathbb{Q(\eta_{5})}$?

In Lorenz's Galois Theory book, there's a problem : Why $\sqrt{15} \notin \mathbb{Q(\eta_{15})}$, where $\eta_{15}$ is a $15$-th primitive root of unity ? But My question is about what it's ...
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Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$.

Is $\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$? Say $L=\mathbf Q(\sqrt 2,\sqrt 3,\sqrt 5)$ and $K=\mathbf Q(\sqrt 2+\sqrt 3+\sqrt 5)$. It is easy to show that ...
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0answers
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Explanation of why a differential field extension must be trancendental

I'm attempting to follow the steps of this proof that $e^{x^2}$ has no antiderivative and there is one step that I'm not quite understanding. They state: If K is a differential field then $K_C=\{r ...
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0answers
38 views

If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
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0answers
35 views

Clarification of Zariski's lemma

In wikipedia Zariski's lemma is the following Let $K$ be a finitely generated $k$-algebra and suppose that $K$ is a field. Then $K$ is a field extension of $k$. But if $M$ be a maximal ideal ...
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1answer
47 views

$X$ is algebraic over $E$

Let $X$ be transcendental over a field $F$, and let $E$ be a subfield of $F(X)$ properly containing $F$. Prove that $X$ is algebraic over $E$. Could we maybe use also the following?? Let $f(x) \in ...
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0answers
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finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
2
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1answer
49 views

Field Extensions and Number of Isomorphisms

The picture above is from Dummit and Foote, Third Edition. Later in the book, we find Clearly, the condition of equality is not necessary, as seen by taking the polynomial $ f(x) = ( x ^{2} ...
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1answer
40 views

Isomorphism of Extensions

Let $\mathbb{L}$ an extension field of $\mathbb{K}$ and $\alpha, \beta\in\mathbb{L}$. If they have the same minimal polynomial than $\mathbb{K}(\alpha)\simeq\mathbb{K}(\beta)$, because if: ...
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1answer
53 views

Degree of field extension $F_{p}(X^p,Y^p) \subset F_{p}(X,Y) $

What is the degree of the following field extension? $$ K=F_{p}(X^p,Y^p) \subset F_{p}(X,Y)=L $$ Because of $ F_{p}(X^p,Y^p) \subset_{(1)} F_{p}(X,Y^p) \subset_{(2)} F_{p}(X,Y) $ and $(1)$ the ...
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0answers
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A $p^n$th root of unity is in $N_{F/k}(F^\times)$ if and only if there is an extension of $F$ which is a cyclic extension of $k$ of degree $p^{n+r}$

Hello, I have to solve this homework but I have completely no idea. please help me, it is very difficult for me. any attempt will be welcomed and appreciated. Conditions : $n, r$ are fixed number ...
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1answer
27 views

Galois extension, simple extension, power

Hello, let $k$ be a field with characteristec zero. and suppose $k$($\alpha$) is Galois extension of $k$. then, is it true $k$($\alpha^3$) = $k$($\alpha$) ? or not? Thank you for your help.
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1answer
20 views

Integral closure of a DVR in finite extension of fraction field

Let $(K,|\cdot|)$ be a complete valued field and let $L$ be a field extension with $[L:K]<\infty$. Let $\mathcal{O}_K$ be the valuation ring in $K$ and let $\mathcal{O}_L$ be the integral closure ...
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1answer
47 views

“Twist” of $\mathbb P^n_K$ through a field automorphism.

This question is closely related to this recent one. Suppose that $s:X\longrightarrow\text{Spec}\, K$ is a variety over $K$ (i.e. a $K$ scheme, separated, proper and geometrically integral) and ...
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223 views

Can we construct $\Bbb C$ without first identifying $\Bbb R$?

Sometimes it is useful to consider $\Bbb C$ as our primitive and identify $\Bbb R$ as a subset of $\Bbb C$. Thus we can define $\Bbb R$ (or at least a set with all of the interesting properties of ...
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1answer
28 views

How do we conclude that $K(a,b) \subseteq \mathcal{A}_{E|K}$?

Let $K \leq E$, $\mathcal{A}_{E|K}=\{a \in E \text{ with } a \text{ algebraic } |K\}$ $K \subseteq \mathcal{A}_{E|K} \subseteq E$ We claim that $\mathcal{A}_{E|K}$ is a field. $a, b \in ...
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2answers
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Adjoining a number to a field

When I studied algebra, we talked about fields such as $\mathbb{Q}[\sqrt{2}]$, the rational numbers with the square root of two adjoined to the field. Structures like these are called field extensions ...
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1answer
95 views

Subfield of $\mathbb{R}$ such that $\Bbb R/K$ is finite.

Is there a field $K \subset \mathbb{R}$ such that $1 < [\mathbb{R} : K] < \infty$? i.e a proper subfield of $\mathbb{R}$ such that the field extension $\mathbb{R}/K$ is finite.
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1answer
32 views

Algebraically independent equivalent conditions

I have some problems to understand the field extensions. Namely, Let $K$ be a field and $E$ its extension. Let $x_1,\ldots ,x_n$ in $E$ and $0<k<n$. Show that TFAE Family $(x_1,...,x_n)$ is ...
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1answer
60 views

Degree of field extension question [closed]

find the degree of the field of extension $\Bbb Q(\sqrt{2},\sqrt[4]{2},\sqrt[8]{2})$ over $\Bbb Q$. 1) 4 2) 8 3) 14 4) 32 I think it is 8.
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55 views

Finding Galois group of $x^6 - 3x^3 + 2$

I'm trying to find the Galois group of $$f(x)= x^6 - 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 - 2)$$ I can see the splitting field must be ...
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1answer
23 views

Maximal algebraically independent subset and transcendence basis

I'm studying transcendence basis and I got stuck with the following problem: Let $K$ be a field and $E$ its extension. Let $S$ be a subset of $E$ such that $E$ is algebraic with respect to $K(S)$. ...
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1answer
25 views

Prove algebraic closure

Let $L/K$ be a field extension such that $L$ is algebraically closed. Show that $\{a\in L\mid[K(a):K]\lt\infty\}$ defines an algebraic closure of $K$. So this is the set of minimal polynomials ...
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1answer
20 views

Finiteness of a simple extension

Here I have two propositions from p.521 on Abstract Algebra written by Dummit Foote. Let $\alpha$ be algebraic over the field $F$ and let $F(\alpha)$ be the field generated by $\alpha$ over $F$. ...