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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Show that any finite extension of $\mathbb{Q}$ is not algebraically closed.

EDIT: Entirely wrong question. I wanted to ask something else. How do I show that any finite extension of $\mathbb{Q}$ is not algebraically closed. In other words, the algebraic closure of ...
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Minimal polynomial: is $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$?

I was wondering about the minimal polynomial of real number $$u=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$$ over field $\mathbb{Q}$. As you can see here, I worked out that $u$ is a root of monic ...
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Proof about Number Fields

It is a known result that if $\alpha$ is an algebraic integer in a number field $K$, i.e. $\alpha \in \mathcal{O}_K$, then its trace and norm are integers. I am looking over a proof of this, which ...
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How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
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+50

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
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1answer
15 views

Dimensions in field extensions

How would I be able to determine $[\Bbb Q(\sqrt{42}, \sqrt{-42}):\Bbb Q]$? So far, I think the dimension might be four as the root equation could be $(x^2 + 42)(x^2 - 42)$.
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Field Extension of Rational Functions

Let $L = F(x)$ be the field of rational functions over a field F. Let $u \in L \backslash F$. Let $K = F(u)$. If u can be written as $\frac{f}{g}$ where $gcd(f,g) = 1$, then prove $[L:K]$ = max {deg ...
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1answer
59 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
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1answer
29 views

Prove every finite extension of subfields of $\mathbb{C}$ is simple

This is a question is from a past paper, and it's only worth 8 marks, and it's really got me. I'm allowed to assume that if $L/K$ is such an extension, that there are $[L:K]$ $K$-embeddings of $L$ ...
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1answer
22 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
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1answer
53 views

Basis for $\mathbb{Q}[\sqrt{8}]$ over $\mathbb{Q}[\sqrt{2}]$

Provided that $x^2-8$ is the minimal polynomial for $\mathbb Q[\sqrt8]$ and $x^2-2$ is minimal for $\mathbb Q[\sqrt 2]$ we should have a basis with four elements. Thus far I know $1$ and $\sqrt 2$ ...
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Dummit and Foote page 526

I'm having trouble with a line of example 2 on page 526. Consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$. generated over $\mathbb{Q}$ by $\sqrt{2}$ and $\sqrt{3}$. Since $\sqrt{3}$ is of degree ...
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1answer
31 views

So-called Artin-Schreier Extension

Let $F$ be a field of characteristic $p$. Let $K$ be a cyclic extension of $F$ of degree $p$. Prove that $K=F(\alpha)$ where $\alpha$ is a root of the polynomial $p(x) = x^{p} - x - a$ for $a \in ...
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30 views

Prove that every finite group occurs as the Galois group of a field extension of the form $F(x_{1}, \dots , x_{n})/F$.

I've seen that every abelian group is a Galois group over $\mathbb{Q}$ for some subfield of a cyclotomic field, but I'm not sure about this more general result.
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How is the Dirichlet unit theorem effectively used to solve diophantine equations

Can someone give me an example or a link to one of its use to solve a diophantine equation ? More precisely : this theorem explains the structure of the units of the ring of integers of a number ...
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1answer
20 views

Binomial formula over an arbitrary field

I'm working on a problem (namely, if $\alpha + \beta$ is algebraic over $F$ then $\alpha$ is algebraic over $F[\beta]$), and the binomial formula appeared. For the problem, I used the fact that, for ...
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1answer
141 views

The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
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25 views

Equal fields and dimension

Let $K \subseteq L$ be fields. Show that K=L if and only if $dim_k L=1$ I know that I will have to show both directions of the implication. I'm very new to the topic of fields so I'm still ...
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33 views

Classifying quadratic extensions of $\mathbb{Q}$

I'm studying Artin's Algebra, and the question says to "Classify quadratic extensions of $\mathbb Q$." What would that look like? A quadratic extension of $\mathbb Q$ is just, for $d$ square-free, ...
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Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
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Field Extensions Identities

I'm working on proving some identities but I need some help clarifying the notation and what exactly each statement is saying. Prove the following identities. (a) $K(A) = QF (K[A])$ (b) $R[A_1 ...
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1answer
51 views

Is it possible that the zeroes of a polynomial form an infinite field?

Let $K/F$ be a finite field extension and suppose that $F$ is infinite. Is it possible to have a nonzero polynomial $p \in K[x_1,...,x_n]$ that vanishes in $F^n$?
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Number of automorphisms

I'm having difficulties with understanding what automorphisms of field extensions are. I have the splitting field $L=\mathbb{Q}(\sqrt[4]3,i)$ of $X^4-3$ over the rationals. Now I have to find ...
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If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?

Defn: A field extension $K/F$ is obtained by adjoining $n$th roots if there is a tower of fields $F=K_1\subset\cdots\subset K_n=K$ such that for each $i$, $K_{i+1}=K_i(\alpha_i)$ and there exists ...
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80 views

Problem in field extension.

I am currently studying the theory of field extension. One of the exercises in my book is causing me a trouble. The problem is Give an example of a field E containing a proper subfield K such that ...
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38 views

Notational Clarification - Abstract Algebra

I'm going through a paper on homomorphic encryption by Smart and Vercauteren entitled "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper. In section 2 of the ...
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1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
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Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a ...
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tower of simple extensions of $\mathbb Z$$_p$ for which the full extension will not be simple

Let $s$ and $t$ be independent variables and let $p$ be a prime. Show that in the tower $\mathbb Z$$_p$$(s^p,t^p)\lt\mathbb Z$$_p$$(s,t^p)\lt\mathbb Z$$_p$$(s,t)$ each step is simple but the full ...
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47 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
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59 views

Construct an embedding

I'm dealing with this problem from the book "Field Theory" (Steven Roman) Suppose $F$ and $E$ are fields and $\sigma : F \rightarrow E $ is an embedding. Construct an extension of $F$ that is ...
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Extension to complex numbers

Is there an extension to the complex numbers in which $zz^* = i$ has a solution? (The star denotes conjugation.) EDIT: I'm mathematically ignorant, but I'm guessing such an extension can't be a ...
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When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
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1answer
65 views

Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
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Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
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Find extension of $\mathbb{Q}$ containing components of eigenvectors of a matrix

Given a matrix $\mathbf{A} \in \mathbb{Z}^{d \times d}$ I need to find an algebraic number $a$ of minimal degree, such that all eigenvalues and eigenvector's coordinates of $\mathbf{A}$ belong to the ...
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78 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
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33 views

Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $ F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
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185 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$

Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ ...
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$\mathbb{Q}(\sqrt[p]{q}) \neq \mathbb{Q}(\sqrt[p]{r})$ for $p,q,r$ primes and $q \neq r$.

Let $p,q$ and $r$ be primes in $\mathbb{Z}$ with $q \neq r$. Let $\sqrt[p]{q}$ denote any root of $x^p-q$ and let $\sqrt[p]{r}$ denote any root of $x^p - r$. I need to prove that ...
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$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}$?

Is there an easy way to see that $$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}?$$ I know that $\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})$ is a subfield of ...
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Infinite field extension but algebraic subset

Is there a subset $ S \subseteq \mathbb{C}$ so that all $s \in S$ are algebraic over $\mathbb{Q}$ and $\left[\mathbb{Q}[S] \colon \mathbb{Q} \right]=\infty$? $\mathbb{Q}[S]$ is defined as ...
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For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of ...
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Showing that $\mathbb{Q}(\zeta_p, \sqrt[p]{\ell}) = \mathbb{Q}(\zeta_p + \sqrt[p]{\ell})$ for $p,\ell$ primes.

We consider the polynomial $x^p - \ell$, where $p,\ell$ are both prime numbers. Let $\zeta_p$ be a $p$-th root of unity. We wish to show that $L = \mathbb{Q}(\zeta_p, \sqrt[p]{\ell})$ is the same as ...
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Why is every monomorphism of E into $\overline{\mathbb{Q}}$ a $\mathbb Q$-monomorphism of E into $\overline{\mathbb{Q}}$?

I've been trying to solve the problem below, but I'm not even sure how to get started. Any help would be greatly appreciated. I feel like there is a key insight that will solve the problem, but I'm ...
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1answer
11 views

Tower rule for transcedence degree

$\newcommand{\tr}{\operatorname{tr}}$ Let $k \subset E \subset K$ be extension fields. Show that $$\tr\ \deg (K/k)= \tr\ \deg (K/E) + \tr\ \deg (E/k).$$ Here if we consider $S_1$ and $S_2$ be two ...
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45 views

Galois Theory: An automorphism fixes a field if and only if it fixes the set of generators.

Let $F/K$ be a field extension. Let $a_{1},...,a_{n}\in{F}$ and $E:=K(a_{1},...,a_{n})$. Then how do we show $\sigma\in{Aut_{E}F}$ if and only if $\sigma(a_{i})=a_{i}$ for all $i=1,2,...,n$? Any ...
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finite field extension problem

Maybe somebody knows how to proove the following algebraic theorem: $C \subset U$ is a field extension and $N \subset U$ so, that all $x \in N$ are algebraic over $C$ and $C[N]=\left\lbrace ...
2
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1answer
48 views

Separable field extensions

Let $k$ be a field and $k(x_1,x_2,...x_n)= k(x)$ a finite separable extension. Let $u_1,u_2,..., u_n$ be algebraically independent over $k$. Let $w= u_1x_1 + u_2 x_2 +\cdots +u_n x_n .$ Let $k_u = ...
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79 views

If $u,v \in K$ and $u+v$ is algebraic over $F$, is it true that $u$ is algebraic over $F(v)$?

Exercise from "Abstract Algebra: An Introduction" by T.W.Hungerford. Let $K$ be an extention field of $F$. If $u,v \in K$ and $u+v$ is algebraic over $F$, prove that $u$ is algebraic over $F(v)$. ...