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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2
votes
4answers
81 views

A question about the definition of $\mathbb{C}$

In the usual definition of the field $\mathbb{C}$, as $\{(a,b):a,b\in\mathbb{R}\}$, the field $\mathbb{R}$ is not exactly a subset of $\mathbb{C}$, but only an isomorphic copy of the subfield $\{(a,0):...
-1
votes
1answer
34 views

Degree of field extension and rational function field

Suppose that $k$ and $k'$ are fields such that $k\subset k'$ and $[k':k]=n$, where $n$ is an positive integer. Do we have $[k'(x):k(x)]=[k':k]$? Why? Thanks for your help!
4
votes
2answers
47 views

Linking two theorems: on algebraic closure and on minimal splitting field

Consider following two statements from same book of Cohn: Basic Algebra. Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting ...
0
votes
1answer
46 views

Existence of proper field extension

I am wondering whether the following statement is true or not? Given any field $F$, there exists a proper field extension $K$ of $F$.
-2
votes
1answer
27 views

Field extension $\mathbb F_p\subset E$ [closed]

Suppose there exists a field extension $\mathbb F_p\subset E$. Question: Is it possible that the degree is $[E:\mathbb F_p]=2$. And how many elemnts are in E then? How can I proof such a question?...
2
votes
2answers
37 views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...
0
votes
1answer
19 views

the field $Fr (A [X])$ and $ Fr (A) (X)$ are the same

Let $A$ be an unitary integral domain , and let $Fr (A)$ its fractionary field; this field is determined (to isomorphism field) by the following universal property: a) the ring $A$ is injected by a ...
1
vote
0answers
48 views

Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
2
votes
0answers
44 views

Automorphism of $\mathbb C(x,y)$ and its order in $\mathrm{Aut}(\mathbb C(x,y))$

In the following problem Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$. (i) Show that $\phi(x)=...
1
vote
3answers
80 views

Quick question about $\mathbb{C}$ considered as field extension of $\mathbb{R}$

In algebra, one way to understand $\mathbb{C}$ is to consider it as a field extension of $\mathbb{R}$. What (sometimes) worries me is this: From this point of view, does $\mathbb{C}$ really contain a ...
1
vote
2answers
57 views

What happens if I take a quotient over a reducible polynomial?

I know that for adjoinging roots to a field, I need to find irreducible polynomials so that the ideal I am taking the quotion with will be maximal, hence the resut being a field. Imagine I am working ...
2
votes
1answer
32 views

Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
2
votes
2answers
116 views

Field extensions and monomorphisms

I am working through some algebra exercises and got stuck with the following problem: I am working in the finite field $\mathbb{Z}_5$. Let $f \in \mathbb{Z}_5[x]$ and $f(x) = x^2 + 2$. By simple ...
4
votes
3answers
90 views

Factorization of polynomials over $\mathbb{Z}_3$

I have been given these two polynomials $$f(t)=t^3+2t+1 \text{ & }g(t)=t^3+t^2-t+2$$ the problem says, decide if both factorization fields are isomorphic. For the second polynomial I got that $$g(...
1
vote
0answers
51 views

Question about the Fundamental Theorem of Galois Theory

This is something more like a small doubt than some problem that I need help with. I'm doing and exercise that is asking me to find subextensions of a given extension of all posible orders, and find ...
0
votes
1answer
35 views

A reduction to the finite degree case

I am stuck trying to understand a proof in Asymptotic Differential Algebra and Model Theory of Transseries by L. van den Dries, J. van der Hoeven and M. Aschenbrenner. The result is the following: ...
1
vote
0answers
58 views

Existence of a division ring on a field.

Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$. In the field extensions we know that $a^2-2ab+b^2=0$ if and ...
0
votes
0answers
23 views

Field extension over the rationals does not have a square root of -$\alpha^2$

Let $f=x^4-2\in\mathbb{Q}[x]$ and consider the field $K=\mathbb{Q}[x]/(f)$. I want to show that There exists no element $u\in K$ such that $u^2=-\alpha^2$, where $\alpha$ is the coset of $x$. ...
1
vote
1answer
47 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(...
6
votes
2answers
147 views

Isomorphic fields of finite degree have same dimension over base field

Let $K/F$ be a field extension and $L_1,L_2$ subfields of $K$ such that $L_1$ and $L_2$ have finite degree over $F$. Does $L_1 \cong L_2$ imply $[L_1 : F ]=[L_2 : F]$? Obviously, if the isomorphism ...
0
votes
2answers
86 views

Proving $\mathbb R[x]/\langle 1+x^2\rangle$ $\cong$ $\mathbb C$ without using 1st isomorphism theorem

I've seen many the proofs of this by making use of First isomorphism theorem, by considering the map,$$\phi:\mathbb R[x]\rightarrow\mathbb C$$ defined by $\phi(a+bx)=a+bi$. My questions are ...
0
votes
2answers
31 views

If a Galois group has $n$ subgroups of some order $k$, will there always be $n$ intermediate field extensions of order $k$?

I realised today that I don't really understand the entirety of the fundamental theorem of Galois theory. It might be that the way it's phrased in my book confuses me, or it might be the subject ...
1
vote
1answer
46 views

Prove there is no such nth root of unity $\zeta$ such that $\mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\zeta) \quad$ [duplicate]

I'm trying to do the above problem. My approach is to use the fact that $\mathbb{Q(\zeta)}$ is the fixed subfield of the normal subgroup $A_3$ of $S_3$ and then since $A_3$ has no subgroup of the form ...
0
votes
1answer
23 views

Tower of fields - Normal [closed]

I need to construct a tower $$k \subseteq K \subseteq L,$$ such that $K/k$ is normal and $L/K$ is normal, but $L/k$ is not normal.
0
votes
1answer
17 views

Knowing the Galois group of the splitting field of a polynomial $f$, how can I show that $f$ is irreducible in the ground field?

So I'm given $f(x) = \sum_{k=0}^{8}\frac{x^k}{k!} \in \mathbb{Q}[x]$. Denote its splitting field by $E$, then I'm also given that ${\rm Gal}(E/\mathbb{Q}) \cong A_8$. The task is to prove that $f(x)$ ...
0
votes
1answer
40 views

Solvable but not radical.

Is there an example of a field extension that is solvable, but not radical? I also would like an example of an extension that is radical but not solvable. Been trying to come up with these examples ...
0
votes
0answers
20 views

Given a tower of extensions how to show that the degree of an extension is even?

Suppose $\Bbb{Q} \subseteq F \subseteq \Bbb{C}$ is a tower of extensions and suppose that $i \in F$. If the extension $\Bbb{Q} \subseteq F$ is finite, show that $[F : \Bbb{Q}]$ is even. What ...
2
votes
3answers
95 views

Prove $2^{1/2}+3^{1/3}$ is irrational using Galois theory.

So, I want to prove that $2^{1/2}+3^{1/3}$ is irrational, and I need to prove it using Galois theory. To start, let's forget about the sum and deal with the individual numbers and $F_1 = \mathbb{Q}(...
1
vote
1answer
44 views

How can I show that the Galois group of $x^p -1$ is abelian?

So $p$ is prime, and we have $f = x^p -1 \in \mathbb{Q}[x]$ with splitting field $E$. I need to show that ${\rm Gal}(E/\mathbb{Q})$ is abelian and of order $p-1$. The splitting field $E$ is $\mathbb{...
0
votes
0answers
20 views

Let $F=K(u)$ where $u$ is transcendental over $K$, prove that it is algebraic over $E$, where $K \subset E \subseteq F$

Let $F=K(u)$ where $u$ is transcendental over $K$. Prove that it is algebraic over $E$, where $K \subset E \subseteq F$. The method I tried for the above question was as follows: Choose $v \in E/K$ ...
5
votes
2answers
117 views

Neat method to show that $\mathbb{Q}(2^{1/3}) \ne \mathbb{Q}(3^{1/3}) $?

I am wondering how to show looking obvious $\mathbb{Q}(2^{\frac{1}{3}}) \ne \mathbb{Q}(3^{\frac{1}{3}}) $? This question has appeared to compute the order of $\text{Gal}(\mathbb{Q}(2^{\frac{1}{3}},3^{...
2
votes
1answer
47 views

Galois Group Isomorphic to $S_3$.

Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree 3. Suppose $f$ has one real root, we want to show that $$\text{Gal}(L/\mathbb{Q}) \cong S_3,$$ where $L$ is the splitting field of $f$. ...
1
vote
1answer
31 views

Field Extension for which Galois correspondence fails [closed]

Find a non-Galois field extension such that the Galois correspondence fails. Can't seem to come up with a nice answer to this.
1
vote
2answers
32 views

Suppose $F$ is a field extension of $\Bbb{Z}_2$ of degree $3$, prove $F$ is finite, what is the size of $F$?

Suppose $F$ is a field extension of $\Bbb{Z}_2$ of degree $3$, prove $F$ is finite, what is the size of $F$? Okay so what we have $[F:\Bbb{Z}_2]=3$ so if we view $F$ as a vector space over $\Bbb{Z}...
0
votes
2answers
45 views

I need help understanding a proof (Kronecker's theorem)

Kronecker's theorem says that if $F$ is a field and $f(x)$ is a non-constant polynomial in $F[x]$, then there exists an extension field $E$ of $F$ in which $f(x)$ has a root. Here's the proof ...
0
votes
1answer
28 views

the splitting field for $x^p - 1$

I need to show that the splitting field for a polynomial of the form $f(x) = x^p - 1$, where $p$ is prime and the coefficients of $f$ are in $\mathbb{Q}$, can be expressed as $\mathbb{Q}(\gamma)$, ...
-1
votes
1answer
30 views

Degree of Splitting Field of $x^{10}-5$ over $\mathbb{Q}$

I've narrowed it down to either $20$ or $40$: $$x^{10}-5=0\iff x^{10}=5e^{2\pi ik}\iff x=5^{1/10}e^{\pi ik/5}, k=0,1,2,3,4$$ One can show that the splitting field is $\mathbb{Q}(5^{1/10},e^{\pi i/5}...
1
vote
2answers
47 views

K is normal over F.

Let K be a field and suppose that $\sigma \in Aut(K)$ has infinite order. Let F be the fixed field of $\sigma$. If K/F is algebraic, show that K is normal over F. Note: $F=\{x \in K| \sigma(x)=x \}$ ...
1
vote
0answers
18 views

Let $K$ be an extencion of $F$ and let $a,b\in K$. If $a$ is not algebraic in $F$, but it is in $F(b)$, show that $b$ is algebraic in $F(a)$ [duplicate]

Let $K$ be an extencion of $F$ and let $a,b\in K$. If $a$ is not algebraic in $F$, but it is in $F(b)$, show that $b$ is algebraic in $F(a)$. $F(x)$ is the smaller subfield of $K$ containing $F$ and ...
3
votes
4answers
116 views

minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...
1
vote
0answers
21 views

Galois correspondence for $\mathbb{Q}(\sqrt[4]{2},i)/\mathbb{Q}$

I've determine that this extension has degree 8 and a basis of this extension is given by $$\{1, i, \sqrt[4]{2}, \sqrt[4]{2}i, \sqrt{2}, \sqrt{2}i, \sqrt[4]{8}, \sqrt[4]{8}i \}.$$ This reveals to us ...
0
votes
0answers
27 views

Find the normal closure of $\mathbb{Q}({\sqrt{-5+2\sqrt{5}}})$

Let $\mathbb{Q}(\sqrt{-5+2\sqrt{5}})$ Find the normal closure, $L$ of $\mathbb{Q}({\sqrt{-5+2\sqrt{5}}})$ $\mathbb{Q}({\sqrt{-5+2\sqrt{5}}}$ )$=\mathbb{Q}(\sqrt{5})(\alpha)$ $L=\mathbb{Q}(\sqrt{...
-1
votes
0answers
30 views

Degree of field extensions $k(\alpha) = k(\alpha^p)$

Let $p$ be a prime and $k$ a field such that $x^p-1$ splits into linear factors. Now suppose that $k \subset K$ is a field extension, and that $\alpha \in K$ has minimal polynomial $f \in k[x]$ of ...
0
votes
1answer
38 views

Splitting field of $x^3-5 \in \mathbb{Q}[X]$. Galois group and fields?

I have this multi-part problem I have worked on in Galois Theory. I am particularly unsure abut finding all roots of our polynomial and the action of the Galois group. Also, I cannot see how we can ...
0
votes
1answer
68 views

Prove there is an $A \in \mathbb{Q}$ such that $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$

Let $K=\mathbb{Q}(\sqrt{-13+2\sqrt{13}})$. $K$ is normal over $\mathbb{Q}$ Prove there is an $A \in \mathbb{Q}$ such that $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$ So we need to show that $K(i)=\mathbb{...
0
votes
1answer
14 views

Generators of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ - what is $\tau(\zeta_n)$?

I am trying to understand the structure of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})\simeq Z_n^*$ for $n \in \mathbb{Z}$ - it would be great if someone could me understand the generators in this group so ...
1
vote
2answers
25 views

Existence of subfields such that $[\mathbb{Q}(\zeta_{25}) : K_1]=4$ and $[\mathbb{Q}(\zeta_{25}) : K_2]=5$

I have the following two questions questions I am working on and am a little stuck : Let $L=\mathbb{Q}(\zeta_{25})$ where $\zeta_{25}$ is the primitive $n$-th root of unity. Prove that there are ...
3
votes
2answers
64 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
18 views

Irreducible polynomial of degree $5$ in $L[X]$ using Kummer's theory

Let $L=\mathbb{F_2}(\theta)$ where $\theta$ is a root of $X^4+X+1$ I am trying to solve the following consecutive questions in Galois Theory: Prove that $L$ has degree $4$ over $\mathbb{F_2}$ ...
2
votes
2answers
29 views

Irreducibility of $X^5-7$ over $\mathbb{Q}(\sqrt[7]{2})[X]$ and degree of spitting field

I have worked through these two questions but am unsure if I got the right idea, please may you help me? Prove that $X^5-7$ is irreducible over $\mathbb{Q}(\sqrt[7]{2})[X]$ Can we say that $f(X)...