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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Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? [on hold]

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? I had $\mathbb{C}$ as a field, $\mathbb{C}(x)$ as a field extension, and $\mathbb{C}[x]$ ...
0
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1answer
23 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 ...
0
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1answer
15 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 $F$. ...
0
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1answer
20 views

Does the fixed field of automorphisms group characterize Galois extensions?

If $E/K$ is a field extension we use the notation $\def\Aut{\operatorname{Aut}}\Aut(E/K)$ for the set of field automorphisms of $E$ that are the identity over $K$. It's immediate that the set ...
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0answers
13 views

Sufficient condition for a polynomial to split

I found a problem that reads: let $K\subseteq L$ be two fields, and consider an irreducible $f(x)$ in $K[x]$. Show that if there exists an $a\in L$ such that $a$ and $a^2$ are roots of $f$, then $f$ ...
3
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2answers
61 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
0
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0answers
28 views

constructibility of a number

Following task: If $\alpha$ and $\beta$ are algebraic numbers, having the same minimal polynomial, then I should show that $\alpha$ is constructible if and only if $\beta$ is constructible. I don't ...
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0answers
19 views

Finding subfields of a field extension [duplicate]

How do I find all subfields of $\mathbb{Q}(e^{\frac{\pi i}{4}})$? I'm very thankful for any help and tips.
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45 views

Factorization and Roots of the following Polynomials

I'm struggling with this exercise $(a)$$f:=T^4 +6T^2 -8T - 3 \in \mathbb{Q}[T]$. Show that f is irreducible, with exactly $2$ real roots. $(b)$ Let $\alpha$ und $\beta$ be two real roots of ...
0
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1answer
19 views

About finite field extensions and their generators

I am with trouble to prove this statment: "Suppose that $\{\beta_1,...,\beta_n\}$ is a base of $L|K$ and $\mathcal{M}$ is a subset of some fied $M\supseteq L$. Prove that $\{\beta_1,...,\beta_n\}$ ...
1
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1answer
69 views

Subfields of $\mathbb Q(e^{\frac{i\pi}{4}})$

What are the subfields of $\mathbb Q(e^{\frac{i\pi}{4}})$ ? Since $\displaystyle e^{\frac{i\pi}{4}}=\frac{\sqrt2+\sqrt{-2}}2$ So $\mathbb Q(\sqrt2), \mathbb Q(\sqrt{-2})$ and ...
1
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1answer
42 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
2
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1answer
33 views

The normal closure of a field extension

I'm making my first steps in abstract algebra and I was wondering, if there is a technique to determine the normal closure of a given extension, cause all I know is a theoretical definition: $K_n$ is ...
0
votes
1answer
13 views

Approximating a field by perfect fields.

Let's consider an arbitrary field $K$ and raise the following question: in which sense can we approximate $K$ by a perfect field? Any reasonable notion of approximation by a perfect field should admit ...
3
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0answers
33 views

A question concerning cyclic field extensions.

In the study of cyclic extensions we have the following theorem: Theorem Let $K$ be a field containing an $n$-th primitive root of unity $\zeta$. Then the following claims hold: If ...
0
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1answer
13 views

Question about field extensions regarding minimal polynomial of multiple of algebraic element

Let $K/F$ be a field extension, let $α ∈ K$ be an algebraic element with minimal polynomial $f(X) ∈ F[X]$, and let $r ∈ F^\times$. What is the minimal polynomial for $rα$ in terms of $r$ and $f$? I ...
1
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1answer
25 views

Question regarding algebraicity of two elements whose sum and product are algebraic.

Let $\alpha , \beta \in \Bbb C$ and suppose $\alpha + \beta$ and $\alpha \beta$ are algebraic over $\Bbb Q$. Prove $\alpha , \beta$ are algebraic over $\Bbb Q$.
3
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1answer
29 views

Field of rational functions

Let $K$ be a field with characteristic $p>0$ and $M=K(X,Y)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K(X^p,Y^p)\subset M$. Show that $[M:L]=p^2$. I ...
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1answer
27 views

Let $F|K$ be a field extension and $a \in F $ such that $[K(a):K]$ is odd integer [duplicate]

Let $F|K$ be a field extension and $a \in F$ such that $[K(a):K]$ is odd integer, then prove that $K(a)=K(a^2)$.
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0answers
7 views

Purely inseparable extension from Hungerford

Hungerford, Algebra, V.6.4 says, $F/K$ is purely inseparable if and only if $F$ is generated by a set of purely inseparable elements over $K$. My question : is there any purely inseparable extension ...
0
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1answer
31 views

Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
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1answer
16 views

How to show $[F(a):F[a^4+a^2+1]$ is finite.

I have an element $a$ in an extension field $F$. I'm asked if it is true that $a^4+a^2+1$ is algebraic over $F$ if and only if $a$ is algebraic over $F$. I know that if I can show ...
2
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1answer
22 views

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Let $F$ be a field and $K$ be an extension field of $F$. Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$. I haven't ...
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1answer
32 views

Quadratic extension of quadratic extensions

I need help for the following exercise: The field $\mathbb{Q}(e^{\frac{2 \pi i}{3}})$ is a quadratic extension of $\mathbb{Q}$ and $\mathbb{Q}(e^{\frac{\pi i}{6}})$ is a quadratic extension of ...
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1answer
31 views

If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Let $a$ and $b$ be elements in extension field $F$. Is it true that: If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$? I just did the same ...
2
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1answer
39 views

Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is ...
4
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1answer
72 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
3
votes
1answer
26 views

Proving $f\in K(\beta)[X]$ irreducible iff $g\in K(\alpha)[X]$ irreducible

Let $K\subset L$ a field extension and $\alpha,\beta\in L$ with minimal polynomials $f,g\in K[X]$. How to show that $f\in K(\beta)[X]$ is irreducible iff $g\in K(\alpha)[X]$ is irreducible?
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55 views

Discriminant of a field extension

I'm struggling with this exercise: Let $K:=\mathbb{Q}(\sqrt{2},\sqrt{-1})$ and $R:=\mathbb{Z}(\sqrt{2},\sqrt{-1})$. a) Compute the discriminant of $R$ over $\mathbb{Z}$. Suppose that $S$ is ...
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1answer
23 views

Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements ...
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2answers
66 views

Showing a counterexample regarding normal extension

For field extensions K/E, E/F, if K/F is a normal extension, E/F is a normal extension also? I think this is false..but can't find a counterexample. Could anyone suggest me some example?
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27 views

the motivation of separable field extension

What is the origin of the motivation of separable field extension? Is it related to separable topological space or something else?
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3answers
89 views

If $F$ is an extension field of the field $K$ such that $[F:K] =1$, then $F=K$

Suppose $F$ is an extension field of the field $K$ such that $[F:K] =1$. How to prove that $F=K$? Thank you for your time and help.
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1answer
31 views

A primitive element for the extension $\mathbb{Q}(\sqrt{7}, \sqrt[5]{2}) / \mathbb{Q}$

I'm trying to represent the field extension $\mathbb{Q}(\sqrt{7}, \sqrt[5]{2}) / \mathbb{Q}$ as a simple extension over $\mathbb{Q}$. I know the degree of the extension is $10$ and a ...
7
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1answer
54 views

how can I find the smallest integer n such that a polynomial divides x^n-1

I have a simple question.. Assume that I have an arbitrary polynomial $f$ in $F_q[x]$. Is there a practical way to find the smallest integer $n$ for which $f$ divides $x^n-1$ ? A small example ...
2
votes
1answer
18 views

Degree and Basis

I'm trying to express the field $\mathbb{Q}(\sqrt{3}, \sqrt[4]{3})$ as a simple extension over $\mathbb{Q}$. I know that $\left[ \mathbb{Q} (\sqrt{3}):\mathbb{Q} \right] =2 $ and $\left[ \mathbb{Q} ...
0
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0answers
38 views

A particular case of Zariski's lemma

I am trying to prove Zariski's lemma in the following case: let $k$ be a field and $k[x]$ be an algebra generated by $x$, such that $k[x]$ is a field. Then the extension $[k(x):k]$ is finite. ...
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0answers
34 views

Zariski's lemma

I have a question concerning Zariski's lemma: is it still true if we assume the ground field to be a finite one? $k \subseteq K$ field extension, with $k$ a finite field (of $p^n$ elements) and $K$ ...
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0answers
35 views

Degree of the field extension for $x^4+3$

I want to find a splitting field for : $x^4+3 \in \mathbb{Q}[x]$ and the degree of that extension , I would be thankful if you verify/correct what I have tried for this problem : First I find the ...
2
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1answer
35 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
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1answer
21 views

Calculate the degree of $[\mathbb Q(a):\mathbb Q]$

How to determine $[\mathbb Q(a):\mathbb Q]$ for $a\in\overline{\mathbb Q}\setminus\mathbb Q$ with $a^p=1$ with $p$ prime? So $a$ is an algebraic, complex number which cannot be written as a quotient. ...
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1answer
26 views

Finite field extensions and minimal polynomial

I want to show the following statement: Let L/K be a finite field extension with $[L:K]=p$ for a prime $p$ Show that $[L:K]$ is simple Proof: 1) Choose $\alpha\in L$ with $\alpha \notin K$. Then ...
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1answer
35 views

How can you write $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ using a single algebraic element $\mathbb{Q}[\alpha]$?

Looking at the basis of $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ gives me no idea on how to generate it using $\{1, \alpha, \alpha^2,\alpha^3,\alpha^4,\alpha^5\}$ for some $\alpha$ algebraic over ...
4
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1answer
58 views

Minimal polynomial over a field

I have some troubles with this exercise: Let $a$ and $b$ be the following elements of $\mathbb{Q}(T)$: $a:=T^3+T$ and $b:=T^2 -2$ Compute the minimalpolynomial of $b$ over $\mathbb{Q}(a)$, and ...
2
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1answer
105 views

Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.

These rules are part of an attempt to define an additive identity in terms of division in basic standard arithmetic. The difficulties with defining division by $0$ are well known. In order to ...
3
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1answer
28 views

Calculating the degree of $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$

Let $m\in\mathbb Z$ with a prime factorization of the form $m=p\Pi p_i^{n_i}$, $p\neq p_i$. How can I calculate $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$ for a natural number $n$?
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1answer
48 views

Integral extensions with finitely generated k-algebras

I have $k$ a field, and I am assuming that the finitely generated $k$-algebra $K = k[x_1,x_2]$ is also a field. I am trying to prove Zariski's lemma in this case, by seeing first that $K$ is an ...
0
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1answer
15 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
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2answers
24 views

Why does it stand that #$\mathbb{Z}_p(a)=p^n$?

If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$ The proof is the following: Let $a \in K$. We take $\mathbb{Z}_p ...
2
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1answer
42 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...