Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic....

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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 $x$. ...
0
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2answers
30 views

Discrete Mathematics (Closure Problems)

$R = \{(x, x+1)|x \in \mathbb{Z}\}$ $\mathbb{Z}$ is the integers and could be negative or positive. Create the closure of the the following: a. $t(R)$ --> transitive closure of R b. $rt(R)$ --> ...
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0answers
48 views

Describe the separable closure of $Z_3(u,v)$ in $Z_3(y,z)$

I'm struggling with the separable closure problem and I don't understand some points. Please explain why it is.. WTS : Describe the separable closure of $Z_3(u,v)$ in $Z_3(y,z)$ Let y,z be ...
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2answers
92 views

Show that $\mathbb{Z}[i]/n\mathbb{Z}[i] $ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$.

I have to show the following statement: $\mathbb{Z}[i]/n\mathbb{Z}[i]$ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$. Let $\mathbb{Z}[i]/n\mathbb{Z}[i]$ ...
1
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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
69 views

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
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0answers
13 views

Each exponent of each term of an irreducible polynomial is divisible by p

I'm studying the field theory,in particular, the separable extension. My question is the followings. WTS : an irreducible polynomial q(x) over a field F of characteristic p≠0 is not separable iff ...
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3answers
170 views

On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
6
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2answers
149 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 ...
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2answers
62 views

Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$, let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. [closed]

Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$. Let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. Then A. $[L,\mathbb{Q}]=10$ B. $ [L,K]=2$ C. $[K,\mathbb{Q}]=4$ D. $L=K$
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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.
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0answers
42 views

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$. Suppose that $f$ is a irreducible polynomial of ...
0
votes
0answers
13 views

Separable extensions are distinguished

I'm studying Steve Roman's book "Field Theory" and I found this proof about separable extensions being distinguished but I don't understand his proof. More exactly, why does he conclude from $F<F(\...
0
votes
1answer
18 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 $f(x)$ ...
3
votes
1answer
42 views

What does $K(A)$ mean in field theory?

So in my notes it says that if $K\subset L$ is a field extension and $A \subset L$ is a subset then $K(A)$ is a subfield of $L$ containing both $K$ and $A$. It is in fact the smallest such subfield. I ...
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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 ...
4
votes
2answers
135 views

Is $\sqrt{7} \in \mathbb{Q}(\sqrt{3+\sqrt{2}})\;$?

Let $u = \sqrt{3+\sqrt{2}}\;$. Is $\sqrt{7} \in \mathbb{Q}\left(u\right)$? Equivalently, is $\mathbb{Q}(u)$ a splitting field of $u$ over $\mathbb{Q}\,$? The original question is whether or not $\...
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
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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^{...
1
vote
1answer
29 views

About separable extensions (one more time)

Well I'm stuck trying to prove the following about separable extensions. If $L/E$ is a extension (not necessarily finite) such that $L/F$ and $F/E$ are both separables, then $L/E$ is also separable. ...
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.
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 ...
6
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0answers
84 views

Abelian groups whose finite subgroups are cyclic

If $(F,+,\times)$ is any field, then the abelian group $(F-\{0\},\times)$ has property that every finite subgroup of it is cyclic. Question: If $G$ is an abelian group such that every finite subgroup ...
5
votes
3answers
41 views

Why if $E$ has characteristic $p$ then $F=\{a\in E:a^{p^n}=a\}$ is a subfield?

We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all ...
-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}...
2
votes
2answers
43 views

Finding the degree of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\mathbb Q$ [duplicate]

I am practicing writing down random algebraic numbers and finding their degrees over $\mathbb Q$ and have fumbled when coming to $\sqrt[3]{2} + \sqrt[3]{3}$. Mathematica tells me the minimal ...
2
votes
1answer
48 views

Solvable Groups

Does there exist a group $G$ such that every subgroup $H$ is solvable, but $G$ is not solvable. I know that if $G$ is solvable, then every subgroup $H$ is solvable, but I want to know if there is a ...
1
vote
1answer
24 views

Separable Extension and Splitting Field

Is every Separable extension a splitting field? Does there exist a counterexample? Also, is there an algebraically closed extension that is not separable?
1
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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: $F=\{x \in K| \sigma(x)=x \}$ ...
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0answers
5 views

polynomial as product of distinct irreducible and separability

F field. $f \in F[x]$. I know if $f$ as product of irreducible is squarefree, then $f$ mayn't be separable if field is not perfect. But what is problem in following proof: Let f as product of ...
4
votes
1answer
46 views

Choose a basis of $\mathbb{F}_q/\mathbb{Z}_p$ to do inverse quickly.

Let $\mathbb{F}_q$ be the finite field with $q$ elements ($q=p^n$, $p$ is a prime). $\mathbb{F}_q$ can be regarded as a linear space over the field $\mathbb{Z}_p$ of dimension $n$. The question is: ...
0
votes
2answers
37 views

Is there a unique homomorphism?

Let $K$ be a finite field of order $q$ and $L$ be a finite extension of $K$. Suppose $\tau$ : $L^{\times} \longrightarrow K^{\times}$ is a homomorphism for which $\tau (a) \tau (b) = \tau(ab)$ for ...
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 ...
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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{...
2
votes
1answer
45 views

Nonabelian Galois Group

Let $f(x)$ be an irreducible polynomial in $\mathbb{Q}[x]$ with both real and nonreal roots. Show that its Galois group is nonabelian. Can the condition that $f$ is irreducible be dropped? If not, ...
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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 ...
2
votes
1answer
32 views

$\text{deg}(f)$ is not divisible by $[L:F]$

I am trying to recall an exam question so I am sorry if this question doesn't make full sense. I think some people would know what the actual wording should be after reading it. $F \subseteq L$ is ...
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 ...
1
vote
1answer
23 views

Show that $Gal(K/L)$ is the intersection of all conjugacy classes of $Gal(K/k)$

Let $K/L$ be Galois extension and $F\subset k\subset L \subset K$ fields such that $L$ is the smallest subfield of $K$ such that $L/F$ is normal. Show that $Gal(K/L)=\bigcap_{\sigma \in Gal(K/F)} \...
0
votes
0answers
86 views

Find the number of subfields of a field of cardinality $2^{100}$ [duplicate]

Find the number of subfields of a field of cardinality $2^{100}$ I want to know whether the answer is $9$. But I need a proper logic of that answer.
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 $\mathbb{F_2}$ ...
1
vote
2answers
51 views

Is $X^5+…+1 \in \mathbb{F_2}[X]$ irreducible?

I am trying to determine if the following polynomials are irreducible in $\mathbb{F_2}[X]$ are irreducible: $f(X)=X^5+X^2+1$ $g(X)=X^5+X^3+1$ There are no linear factors since $f(0)=f(1)=g(...
0
votes
2answers
43 views

Non-separable, infinite field extensions of non-zero characteristic

I have been trying to find examples (and non-examples) of fields which are separable, finite and have characteristic equal to zero. Separable Example: $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ because the ...