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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-1
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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
44 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
49 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
vote
2answers
60 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: $F=\{x \in K| \sigma(x)=x \}$ ...
0
votes
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
40 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
120 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
22 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 ...
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, ...
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
40 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
97 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
44 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
votes
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
30 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
32 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 ...
0
votes
0answers
60 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?
0
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
12 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
24 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
79 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
56 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
63 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 $n=8$, ...
2
votes
5answers
96 views

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$

Prove that the $7$-th cyclotomic extension $\mathbb{Q}(\zeta_7)$ contains $\sqrt{-7}$ I thought that the definition of the $n$-th cyclotomic extension was: $\mathbb{Q}(\zeta_n)=\{\mathbb{Q}, \sqrt{-...
2
votes
3answers
68 views

Degree of field extension $\mathbb{Q}\subseteq\mathbb{Q}(i,i\sqrt2)$

I have a field extension $\mathbb{Q}\subseteq\mathbb{Q}(i,i\sqrt2)$ that I want to find the degree of. Usually I find it easiest to find the minimal polynomial, but I can't start by saying $x=i,i\...
0
votes
2answers
51 views

Working with C++ for GF(2) [closed]

Pardon me if it is off topic.But, is there anyone who could suggest me some basics with how to get started with working with C++ for GF(2)?? I am new in C++.I am learning to working with arrays and ...
0
votes
1answer
21 views

Given $F \subset L \subset K$ where $K$ is a Galois ext. of $F$, find an example where $F \subset L$ is not a Galois ext.

I have already shown that if $F\subset K$ is a Galois extension, then for any intermediate field $L$, we have $L\subset K$ is a Galois extension. I then want to show that it's not necessarily true ...
0
votes
2answers
47 views

$\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \mathbb{Q}]=3$

Suppose $p$ is a prime number, $p\equiv1$ mod $3$ and $\mathbb{Q}(\zeta_p)$ is the $p$-th cyclotomic extension. Prove that $\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \...
1
vote
1answer
23 views

minimal polynomial of $\alpha - \beta$ from the minimal polynomial of $\alpha$

If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$. I've been playing around with the minimal polynomial ...
0
votes
1answer
45 views

Minimal polynomial using Galois theory

I have a couple of questions, given below, about the following problem from a course in Galois Theory. Let $K=\mathbb{Q}(\zeta_{13})$. $K$ contains a unique subfield $L_4$ such that $[L_4 : \mathbb{Q}...
0
votes
1answer
15 views

Generator of $Gal(K/\mathbb{Q})$

Let $K=\mathbb{Q}(\zeta_5)$. Prove that there is a $\tau \in G$ such that $\tau \zeta_5=\zeta_5^2$ is a generator of $Gal(K/\mathbb{Q})$ I belive we must consider $\mathbb{Z_5}$, but I am not ...
2
votes
1answer
47 views

If $x^3+px+q$ is irreducible over a finite field then $-4p^3-27q^2$ is a square

Suppose that $x^3+px+q$ is irreducible over a finite field $F$ with characteristic not equal to $2$ or $3$. Show that $-4p^3-27q^2$ is a square in $F$. I noticed that the determinant of $f=x^3+px+q$ ...
0
votes
1answer
32 views

Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$ $L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$ Describe the structure ...
0
votes
2answers
36 views

The degree of an algebraic element over a field extension

Let $ L/K $ be a field extension and let $ \alpha $ be an algebraic element of prime degree over $ K $, i.e $ [K(\alpha) : K] = p $ for some prime $ p $. Is it always the case that we have $ [L(\alpha)...
1
vote
3answers
43 views

$\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$

Let $\alpha$ be a zero of $f(x)=x^3+x+1 \in \mathbb{F_2}[x]$. Show that $\mathbb{F_2}(\alpha)/\mathbb{F_2}$ is a splitting field of $f(x)$ over $\mathbb{F_2}$ So we need to show that $\mathbb{...
3
votes
2answers
40 views

I don't understand this argument about a certain Galois group.

So I'm working with $\alpha = \sqrt{5+\sqrt{5}}$ and $E=\mathbb{Q}(\alpha)$. The minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $f(x) = x^4 -10x^2 +20$ and I've determined that $E$ is its ...
3
votes
2answers
93 views

Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
3
votes
2answers
57 views

Prove that $\mathbb{Q}$ has extensions of any finite degree in $\mathbb{C}$

This is a question from a course in Galois Theory and I am quite confused. In general, the degree of a field extension $E/F$ is the dimension of the vector space $E$. What would $E$ and $F$ be in ...
1
vote
2answers
28 views

Is $Q(2^{\frac14},\sqrt7)/Q(\sqrt2)$ normal?

I think it is because $(x^2-\sqrt2)(x^2-7)$ is a polynomial over $Q(\sqrt2)$ and $Q(2^{\frac14},\sqrt7)$ is the splitting field of this over $Q(\sqrt2)$ $\iff$ $Q(2^{\frac14},\sqrt7)$ is normal. So ...
2
votes
1answer
25 views

I want to show how many intermediate fields there are between $GF(3^{12})$ and $GF(3^4)$.

So $E=GF(3^{12})$ is an extension of $F=GF(3^4)$, and I wish to know how many intermediate fields $F \subsetneq K \subsetneq E$ there are. By a result in Escofier's Galois Theory I have that $G={\rm ...
0
votes
1answer
36 views

$G$ has normal subgroup of order 5

Let $L$ be the splitting field of $x^5-7$ over $Q$ and let $G=\text{Gal}(L/Q)$ (I) Prove that $G$ has a normal subgroup of order $5$ (II) Prove that $G$ has a subgroup of order $4$ that is not ...
0
votes
1answer
27 views

Find $[\mathbb Q(\sqrt3, \sqrt{5},2^{\frac13}):Q]$

$[Q(\sqrt3, \sqrt5:Q]=4$ $$ \begin{matrix} & & \mathbb Q(\sqrt3, \sqrt{5})(2^{\frac13}) & & \\ & \stackrel{a}{\diagup} & & \stackrel{b}{\diagdown} \\ \mathbb Q(\sqrt3, \...
2
votes
3answers
46 views

How can I prove that $\mathbb Z_3[i]\cong \mathbb Z_3[x]/\langle x^2+1\rangle$?

Question: Let $\mathbb Z_{3}\left [ i \right ]=\left \{ a+bi\mid a,b \in \mathbb{Z}_{3} \right \}.$ Show that the field $\mathbb Z_{3}\left [ i \right ]$ is ring isomorphic to the field $\...