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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.

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 ...
44
votes
5answers
5k views

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 ...
7
votes
3answers
1k views

Ring Inside an Algebraic Field Extension [duplicate]

Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
9
votes
1answer
385 views

Finiteness of the Algebraic Closure

Let $\mathbb R$ be the field of real numbers. Its algebraic closure, the field $\mathbb C$, is a finite extension of $\mathbb R$, which has degree 2. Are there other examples of fields (not ...
8
votes
3answers
640 views

Algebraic field extensions: Why $k(\alpha)=k[\alpha]$.

If $K$ and $k$ are fields, $K\supset k$ is a field extension and $\alpha \in K$ is algebraic over $k$, then we denote by $k[\alpha]$ the set of elements of $K$ which can be obtained as polynomial ...
17
votes
1answer
478 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 ...
6
votes
3answers
2k views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
5
votes
1answer
511 views

A basis for $k(X)$ regarded as a vector space over $k$

Can anyone give an explicit basis of the $k$-vector space $k(X) = \operatorname{Quot}(k[X])$ of rational functions over $k$? The dimension is given by $$\dim_k k(X) = \max(|k|, |\mathbb N|).$$ If $...
3
votes
3answers
4k views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be $1,(\...
4
votes
1answer
215 views

Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field (...
6
votes
1answer
486 views

Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers. [duplicate]

Take $p_1, p_2, \ldots, p_n, p_{n+1}$ be $n+1$ prime numbers in $\mathbb{P} \subseteq \mathbb{N}$. $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ seems to be quite ...
11
votes
3answers
648 views

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 $\mathbb{F}_p((X))$?
6
votes
4answers
735 views

Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing

I need to rationalize $\displaystyle\frac{1}{a+b\sqrt[3]2 + c(\sqrt[3]2)^2}$ I'm given what I need to rationalize it, namely $\displaystyle\frac{(a^2-2bc)+(-ab+2c^2)\sqrt[3]2+(b^2-ac)\sqrt[3]2^2}{(a^...
3
votes
1answer
468 views

Embedding Fields in Matrix Rings

Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb ...
16
votes
3answers
1k views

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 $F(\sqrt{2})={\Bbb R}$ ...
8
votes
3answers
462 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
9
votes
5answers
965 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 ...
2
votes
1answer
559 views

Intermediate fields of a finite field extension that is not separable

Let $\mathbb{F}_p$ be the finite field with $p$ elements, where $p$ is a prime number. Let $x$ and $y$ be transcendental and algebraically independent over $\mathbb{F}_p$. The extension $\mathbb{F}_p(...
8
votes
4answers
553 views

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?

Is there a general or elegant way to approach this problem? One can show that $\sqrt 5+\sqrt[3] 2$ is a root of the hexic $x^6-6x^4-10x^3+12x^2-60x+17$, which should then be its minimal polynomial to ...
7
votes
3answers
2k views

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group [duplicate]

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 \...
3
votes
3answers
395 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} ...
3
votes
1answer
898 views

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$

Calculate the degree of the extension $[\mathbb{Q}(\cos(\frac{2\pi}{p})):\mathbb{Q}]$ where $p$ is a prime number. My thoughts are: I am lost My intuition says it has to be $ \frac{p-1}{2}$ and that ...
6
votes
2answers
2k views

Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial....
1
vote
1answer
1k views

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

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$. Here $[F:K]$ denotes the dimension of $F$ over $K$ as a vector ...
10
votes
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 $\...
4
votes
2answers
993 views

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

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 ...
2
votes
1answer
121 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
1
vote
1answer
385 views

Extension fields of coprime degrees

Let $u,v$ be algebraic over the field $K$, such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m,n)=1$, then $[K(u,v):K]=mn$. Once $[K(u,v):K]=[K(u,v):K(u)][K(u):K]=n[K(u,v):K(u)]$, I'm ...
1
vote
2answers
153 views

Finite Galois extensions of the form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$

All the extensions with this form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$ are Galois extensions? where $p$ is prime, $p(x)$ is irreducible and the extension is finite.
3
votes
2answers
474 views

Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?

I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$. I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
2
votes
3answers
429 views

Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$

I'm having trouble finding a basis for $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$. So far I know that $[K=\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2$ and $[L=\mathbb{Q}(\sqrt{2}, \sqrt{3}):K] = 2$, but it's ...
17
votes
2answers
1k views

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 ...
8
votes
1answer
255 views

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Suppose $K$ is a field and $F$ is an algebraic extension of some degree $n=(F:K)$. It is stated that the field of rational functions $F(x)$ is in fact an algebraic extension of the field $K(x)$ and ...
7
votes
2answers
4k views

Extensions of degree two are Galois Extensions.

This question from Allan Clark's "Elements of Abstract Algebra" Show that an extension of degree 2 is Galois except possibly when the characteristic is 2. What is the case when the characteristic is ...
6
votes
2answers
2k views

Determine the degree of the splitting field for $f(x)=x^{15}-1$.

Let $f(x)= x^{15} - 1$. Let $L$ be the splitting field of $f(x)$ over the field $K$. Determine the extension degree $[L:K]$ in each case. a) $K= \Bbb{R}$ b) $K= \Bbb{Q}$ c) $K = \Bbb{F}...
7
votes
1answer
635 views

How can we prove $\mathbb{Q}(\sqrt 2, \sqrt 3, … , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + … + \sqrt n )$ [duplicate]

I want to prove this statement. $$\mathbb{Q}(\sqrt 2, \sqrt 3, ..... , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + .... + \sqrt n )$$ for any $n >1$. It looks like a very hard problem. How ...
9
votes
3answers
1k views

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{...
11
votes
3answers
2k views

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 ...
6
votes
2answers
874 views

degree of a field extension

Let $\alpha$ be a root of $x^3+3x-1$ and $\beta$ be a root of $x^3-x+2$. What is the degree of $\mathbb{Q}(\alpha^2+\beta)$ over $\mathbb{Q}$? My guess is 9, because i found a monic polynomial of ...
5
votes
1answer
219 views

Proof that $K\otimes_F L$ is not noetherian

Let $F$ be a field and $K$ and $L$ be extension fields of $F$ such that $\mathrm{tr.deg}_F(K) = \infty$ and $\mathrm{tr.deg}_F(L) = \infty$. It seems to be proved that $K\otimes_F L$ is not ...
2
votes
4answers
662 views

Simple extension of $\mathbb{Q} (\sqrt[4]{2},i)$

I know the simple extension of $\mathbb{Q} (\sqrt[4]{2},i)$ is $\mathbb{Q} (\sqrt[4]{2}+i)$, but how do I show this? One direction is easy, but I have trouble with the other direction.
4
votes
2answers
231 views

How to show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$

How can I show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$, for distinct primes $p,q?$ The other inclusion is trivial. I tried saying $$(\sqrt{p}+\sqrt{q})^{-1} = \...
0
votes
1answer
300 views

Suppose $\gcd(\deg(f),\deg (g))=1$. Show that $g(x)$ is irreducible in $k(\alpha)[X]$.

This is an assignment. There are two related (I think) problems. Please solve one of them and I will try to solve the other. Let $\alpha, \beta $ be algebraic over $k$ whose irreducible polynomials ...
12
votes
1answer
268 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 $1.$...
7
votes
1answer
897 views

Transitivity of Algebraic Field Extensions

Consider the fields $F, E,$ and $K$, where $F \subseteq E \subseteq K$. If $E$ is algebraic over $F$, and $K$ is algebraic over $E$, show that $K$ must be algebraic over $F$. I know this is a well-...
3
votes
3answers
368 views

Minimal polynomial of $\sqrt[3]{2} + \sqrt{3}$

Suppose I want to find the minimal polynomial of the number $\sqrt[3]{2} + \sqrt{3}$. Now that means I want to find a unique polynomial that is irreducible over $\Bbb Q$ such that $f(x)=0$. Now I ...
1
vote
2answers
291 views

How many quadratic extension are there on a field?

Given a field $F$, how many non-isomorphic quzdratic extension of $F$ are there ? I don't know if there is a general answer, for instance there is only one for $F=\mathbf{R}$, viz. $\mathbf{C}$, and ...
1
vote
1answer
124 views

Convert from a field extension to an elementary field extension

I have a basic question about algebraic field extensions: How can I convert a multiple extension like $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to a single (elementary) field extension (like $\mathbb{Q}(\sqrt{...
3
votes
1answer
226 views

Can we extend the real numbers by using hexagonal coordinates on a plane?

What if we used three axes on a plane like in the picture below? Then we could define any point on a plane using three numbers: $$P=(a,b,c)$$ Three numbers seems excessive, however we do not need to ...
2
votes
1answer
79 views

Is $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ a Galois extension of $\mathbb{Q}$

From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question. Would it be right to use ...