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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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
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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 $f(x)$ ...
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 ...
0
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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
132 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
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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
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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
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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
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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.
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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
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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 ...
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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
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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
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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
45 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
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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
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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
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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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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)} \...
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votes
0answers
85 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
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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
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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
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2answers
42 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 ...
0
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0answers
36 views

Prove that a field extension is normal and find the Galois group

Let $K=\mathbb{Q}(\sqrt{2},\sqrt{3},a)$ where $a^2= (9-5\sqrt{3})(2-\sqrt{2})$. Prove that $K/\mathbb{Q}$ is normal and find the Galois group $Gal(K/\mathbb{Q})$. First, I need to find a polynomial ...
1
vote
1answer
28 views

Number of fields between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_n)$

If $\zeta_n$ is the $n$-th primitive root of unity then $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq Z_n^*$ due to the following map $$\tau(\zeta_n)=\zeta^n$$ I was wondering if we could use this ...
1
vote
2answers
31 views

Roots of unity of quadratic extensions of $\mathbb{Q}$.

I am struggling with finding all roots of unity in $\mathbb{Q}(i)$. I know that if $a+bi$ is a root of unity in $\mathbb{Q}(i)$, then $a^2+b^2=1$, and I know how to find all $a, b \in \mathbb{Q}$ that ...
2
votes
1answer
17 views

Transcendental Extensions are of the type $K(x_{1}, \ldots , x_{n})$?

Let $K \mid L$ be a transcendental field extension. Even more, suppose that the transcendental degree of $K$ over $L$ is $n$. Is it true that we can find $x_{1}, \ldots, x_{n} \in K$ in such a manner ...
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votes
0answers
53 views

Show that $\mathbb{Q}[X]/(X^2 + X + 1)$ and $\mathbb{Q}[X]/(X^2 + 1)$ are not isomorphic [duplicate]

How do I show that $\mathbb{Q}[X]/(X^2 + X + 1)$ and $\mathbb{Q}[X]/(X^2 + 1)$ are not isomorphic?
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votes
0answers
15 views

Existence of separable extensions of degree a power of the characteristic

Let $K$ be a field of characteristic $p$. For which $K$ there exist separable extensions of degree $p^n$ for every $n$? Attempt: If $K$ does not contain the algebraic closure of $\mathbb{F}_p$ the ...
2
votes
2answers
48 views

Show that $\alpha = 1 + \sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$; find $\text{Irr}(\alpha:\mathbb{Q})$. [closed]

Show that $\alpha = 1 + \sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$; find $\text{Irr}(\alpha:\mathbb{Q})$. Somehow, 'I can understand' the definition of algebraic elements, but I'm not ...
1
vote
1answer
11 views

Trace of a product of two elements of integral basis

I am struggling with the idea of the trace of $b_ib_j$ where $b_1, \dots, b_n$ form an integral basis of some algebraic number field $K$. I know the trace is the trace of the linear combination of $...
1
vote
0answers
20 views

Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
1
vote
2answers
69 views

Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
1
vote
2answers
55 views

Prove that $K$ is normal over $\mathbb{Q}$ and $K(i)=\mathbb{Q}(i, \sqrt[4]{A})$

Let $K=\mathbb{Q}(\sqrt{-13+2\sqrt{13}})$ Prove that $K$ is normal over $\mathbb{Q}$ Need to show that $K$ is a splitting field of some polynomials in $\mathbb{Q}[X]$. Let $X=\sqrt{-13+2\sqrt{13}...
1
vote
3answers
60 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...