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Questions tagged [extension-field]

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, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.

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Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then ...
38
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4answers
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Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem: If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
32
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3answers
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Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the ...
30
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3answers
513 views

$f(x) $ be the minimal polynomial of $a$ (algebraic element) over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)$ , then is $\mathbb Q(a)=\mathbb Q(b)$?

Let $a \in \mathbb C$ be algebraic over $\mathbb Q$ , let $f(x) \in \mathbb Q[x]$ be the minimal polynomial of $a$ over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)=\mathbb Q[a]$ , then is it true that ...
28
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3answers
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Does every infinite field contain a countably infinite subfield?

Does every infinite field contain a countably infinite subfield? It's easy to see that every field $K$ contains either the rational numbers $\Bbb Q$ (when $K$ has characteristic $0$) or a finite ...
27
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1answer
529 views

Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?

Assume that $p(x)\in \Bbb{Q}[x]$ is irreducible of degree $n\ge3$. Is it possible that $p(x)$ has three distinct zeros $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1-\alpha_2=\alpha_2-\alpha_3$? ...
26
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3answers
649 views

Do extension fields always belong to a bigger field?

Let $F$ be a field, $E_1$ and $E_2$ are two distinct extension fields of $F$. Is it the case that we can always somehow find a field $G$ that contains both $E_1$ and $E_2$? In other words, could ...
19
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2answers
473 views

Without the Axiom of Choice, does every infinite field contain a countably infinite subfield?

Earlier today I asked whether every infinite field contains a countably infinite subfield. That question quickly received several positive answers, but the question of whether those answers use the ...
17
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2answers
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Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ cannot be closed under scalar multiplication by $a \ne 0,1$

The following is the problem 206 from Golan's book Linear Algebra a Beginning Graduate Student Ought to Know. I've been unable to make any progress. Definition: A Hamel basis is a (necessarily ...
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2answers
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Is $\mathbb Q(\sqrt{2},\sqrt{3},\sqrt{5})=\mathbb Q(\sqrt{2}+\sqrt{3}+\sqrt{5})$. [duplicate]

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 $\mathbf Q(\...
17
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1answer
477 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
16
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3answers
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Do maximal proper subfields of the real numbers exist?

To clarify the problem, consider the field ${\Bbb R}$ as a field extension of ${\Bbb Q}$ using some sort of Hamel basis. Does there exist a field $F\subsetneq{\Bbb R}$ such that $F(\sqrt{2})={\Bbb R}$ ...
14
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2answers
422 views

Does there exist a Hamel basis $\mathcal B$ for $\mathbb R$ over $\mathbb Q$ such that $a,b \in \mathcal B \implies \dfrac ab \in \mathcal B$?

This is part of an attempt to understand what multiplicative structure a Hamel basis of the reals over the rationals can have. Does there exist a Hamel basis $\mathcal B$ for $\mathbb R$ over $\...
13
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3answers
291 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 $\...
13
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1answer
339 views

Can we always find a primitive element that is a square?

Let $L/\mathbb Q$ be a finite field extension. The Primitive Element Theorem says that there is an element $\alpha \in L$ so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$ so ...
13
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1answer
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Normal Basis Theorem Proof

I am a little confused by the proof of the Normal Basis Theorem in E. Artin's Galois Theory. Specifically, I am having trouble understanding why a certain squared matrix has a particular form. The ...
13
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1answer
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Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
13
votes
1answer
174 views

Algebraic extension of $\Bbb Q$ with exactly one extension of given degree $n$

Let $n \geq 2$ be any integer. Is there an algebraic extension $F_n$ of $\Bbb Q$ such that $F_n$ has exactly one field extension $K/F_n$ of degree $n$? Here I mean "exactly one" in a strict sense, i....
13
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0answers
234 views

The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ ...
12
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5answers
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The real numbers are a field extension of the rationals?

In preparing for an upcoming course in field theory I am reading a Wikipedia article on field extensions. It states that the complex numbers are a field extension of the reals. I understand this ...
12
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1answer
267 views

Isomorphism between $\Bbb R$ and $\Bbb R(X)$?

My questions are: $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ? $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields? $ $ $ $ For $1.$...
12
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1answer
967 views

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
12
votes
0answers
306 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
11
votes
5answers
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Is $\sqrt[3]{5}$ in $\mathbb Q(\sqrt[3]{2})$? [duplicate]

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 ...
11
votes
2answers
337 views

This tower of fields is being ridiculous

Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2). Now obviously $[\mathbf{C}:\mathbf{R}]=2$...
11
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4answers
187 views

What is the probability that a rational prime remains prime in $\mathbb Z[i,\sqrt{-3}]$?

Using Chebotarev's density theorem, asymptotically, what is the probability that a rational prime remains prime in $\mathbb Z[i, \sqrt{-3}]$?
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3answers
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Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?

We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
11
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1answer
213 views

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

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.
11
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3answers
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A finite field extension of $\mathbb R$ is either $\mathbb R$ or isomorphic to $\mathbb C$

Let $F$ be a field containing $\mathbb R$ with the property that $\dim_{\mathbb R}F < \infty.$ Then either $F \cong \mathbb R$ or $F \cong \mathbb C.$ I am trying to prove the above statement. I ...
11
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1answer
653 views

Geometric interpretation of different types of field extensions?

In a first course on rings and fields we met the concept of field extensions, especially algebraic ones. The presentation of the material was very algebraic and felt a little lifeless. I was wondering ...
11
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1answer
982 views

Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is separable, ...
11
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1answer
262 views

Polynomials $P(x)\in k[x]$ satisfying condition $P(x^2)=P(-x)P(x)$

This question is inspired by this thread which is on hold at the moment. Fix a field $k$. Let $P(x)\in k[x]$ be such that $$(1)\ \ \ \ \ P(x^2)=P(-x)P(x).$$ Let $T(k,d)\subseteq k[x]$ denote the set ...
11
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0answers
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Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$. What is the intersection $F_\infty\cap K_\infty$? (Here $\zeta_{2^n}$ is a ...
10
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4answers
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“Numbers” bigger than every natural number

In the book Understanding analysis, by Abbot, when discussing the Archimedean property, the author states that there are ordered field extensions of $\mathbb{Q}$ that include "numbers" bigger than ...
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3answers
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Why do people study algebraic extension?

Yesterday, I learned Kronecker’s theorem and a finite extension. And now I’m studying the next chapter, Algebraic extension. I think the next theorem shows how important algebraic extension is ...
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4answers
348 views

Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$

I want to show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$. I think it would be easier to prove it using the following: $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt[4]{3})$. Then $\...
10
votes
3answers
308 views

Is $\Bbb Q(\sqrt 2, e)$ a simple extension of $\Bbb Q$?

My general question is to find, if this is possible, two real numbers $a,b$ such that $K=\Bbb Q(a,b)$ is not a simple extension of $\Bbb Q$. $\newcommand{\Q}{\Bbb Q}$ Of course $a$ and $b$ can't ...
10
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4answers
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How do extension fields implement $>, <$ comparisons?

I'm taking an abstract algebra course, and we just hit extension fields - for example, you define $\sqrt{2}$ by starting with the field $\mathbb{Q}$ and defining $\sqrt{2}$ as a solution to the ...
10
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2answers
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Degree of $\sqrt{2}+\sqrt[3]{5}$ over $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{5})$

I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, ...
10
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1answer
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Why are $i$ and $-i$ “more indistinguishable” than $\sqrt{2}$ and $-\sqrt{2}$?

Today I learned that two roots of an irreducible polynomial are "algebraically indistinguishable." In $\mathbb{Q}(\sqrt{2})$, define the conjugate of $a+b\sqrt{2}$ as $\overline{a+b\sqrt{2}} = a - b\...
10
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1answer
292 views

Extension of isomorphism of fields

I'm reading the book of Razmyslov "Identities of Algebras and their representations" and he uses some "supposedly known fact" from field theory. As I could understand, a more or less general statement ...
10
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2answers
117 views

Showing $[K(x):K(\frac{x^5}{1+x})]=5$?

Let $K$ be a field and $x$ be trnacendental over $K$. Compute $[K(x):K(\frac{x^5}{1+x})]$. I've never came across questions like these. It's easy to see that this degree is at most $5$, since: $$x^...
10
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1answer
210 views

Set of elements of degree $2^n$ over a base field is itself a field

Let $F \subset L$ be two fields, and define $K = \{\alpha \in L\mid [F(\alpha): F] \text{ is a power of 2} \}$. Our problem is to prove that $K$ is a field. Closure under reciprocation is easy (...
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0answers
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On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
9
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5answers
964 views

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$ [duplicate]

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
9
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6answers
473 views

Galois Theory by Rotman, Exercise 60, a field of four elements by using Kroenecker's theorem and adjoining a root of $x^4-x$ to $\Bbb Z_2$

I've been working through Rotman's Galois Theory and am stumped by exercise 60: Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4 - x$ to $\mathbb{Z}...
9
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5answers
254 views

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I mean, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ can be viewed as: $\mathbb{Q}[\sqrt{2}][\sqrt{3}]$, as polynomials of $\sqrt{3}$...
9
votes
6answers
2k views

Prove that $x^3-2$ and $x^3-3$ are irreducible over $\Bbb{Q}(i)$

Let $F=\Bbb{Q}(i)$. Prove that $x^3-2$ and $x^3-3$ are irreducible over $F$. How do I go about this? Should I just say that the roots of $x^3-2$ are $2^{1/3},2^{1/3}e^{i2\pi/3},2^{1/3}e^{i4\pi/3}$, ...
9
votes
3answers
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The degree of $\sqrt{2} + \sqrt[3]{5}$ over $\mathbb Q$

I know that the degree is at most $6$, since $\sqrt{2} + \sqrt[3]{5} \in \mathbb Q(\sqrt{2}, \sqrt[3]{5})$, which has degree $6$ over $\mathbb Q$. I'm trying to construct a polynomial with root $\sqrt{...
9
votes
4answers
938 views

Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$. i) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ...