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Questions tagged [galois-theory]

Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-connections).

3
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58 views

Question about Lang's Chapter 6 Theorem 9.1

I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields. He begins his argument with a lemma, If $q \equiv 1$ mod $l$ for ...
2
votes
1answer
32 views

What is the required homomorphism satisfying $f(c)=c$ for all $c\in R$ and $f(X)=aX+b$?

My question is related to this post. I know from the Proposition that Let $φ : R → R'$ be a ring homomorphism. Given elements $a_1, · · · , a_n ∈ R'$ , there is a unique homomorphism $Φ : R[x_1, ...
2
votes
2answers
41 views

Example of a Galois extension $L/\mathbb{Q}$ with $\text{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_{4}$

I propose that $L=\mathbb{Q}(i\sqrt[4]{2})$. Obviously $\mathbb{Q}(i\sqrt[4]{2})$ is the splitting field of $f=t^{4}-2\in\mathbb{Q}[t]$, since $N:=\{\text{zeros of $f$}\}=\{\sqrt[4]{2},-\sqrt[4]{2},...
1
vote
1answer
66 views

How to show that there are infinitely many prime numbers p such that the polynomial f has a zero in Zp? [duplicate]

Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ? I have no ...
7
votes
0answers
80 views

Sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$

What are some sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$ with $\alpha , \beta $ algebraic over $\mathbb{Q}$? We know that, for ...
0
votes
0answers
21 views

How to show that every element involving $x$ in $F_p(x)/F_p$ is not algebraic.

In the field extension $F_p(x)/F_p$, where $F_p(x)$ is the field of fractions of polynomials over $F_p$, is it by definition that $x \in F_p(x)$ is not algebraic? In other words, should I claim that $...
0
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0answers
55 views
+50

If $K/F$ is normal then $K/I$ is Galois.

Let $K/F$ be a normal extension and $I$ be the inseparable closure of $F$ in $K.$ Let $G=\text{Aut}_F(K),$ i.e., $F$ isomorphisms on $K$and similarly define $H=\text{Aut}_I(K).$ Now I have already ...
1
vote
1answer
25 views

Extension of automorphism of field

Let $F$ be a field of characteristic zero, $\overline{F}$ be the algebraic closure of $F$. Let $\zeta_n$ be a primitive $n$-th root of unity in $\overline{F}$. Then it is well-known that $F(\zeta_n)$ ...
1
vote
0answers
37 views

Elements of $\mathbb{Q}(\zeta_p)$ fixed under $\zeta_p \mapsto \zeta_p^g$

I'm reading Harold Edwards’ Galois theory and before going into Galois theory he was discussing about Gauss' works on constructible $n$ gons (so please avoid Galois Theory while answering this ...
0
votes
2answers
34 views

If $\sigma^{*}\left(p\left(x\right)\right)$ is irreducible in $S\left[x\right]$, then $p\left(x\right)$ is irreducible in $R\left[x\right]$.

Let $\sigma:R\to S$ be a ring homomorphism, and $\sigma^{*}$ be the induced map $\sigma^{*}:R\left[x\right]\to S\left[x\right]$ given by $\sum a_{i}x^{i}\mapsto\sum\sigma\left(a_{i}\right)x^{i}$ (this ...
0
votes
1answer
29 views

A quadratic cyclotomic extension

Let $\zeta_n$ be a primitive $n$-th root of unity with $n > 2$. How to show (supposedly using Galois theory) that the extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}(\zeta_n+\zeta_n^{-1})$ has degree $2$...
2
votes
0answers
57 views

Quadratic polynomial satisfied by $\zeta_5+\zeta_5^{-1}$

I got one problem from Dummit Foote stating that determine the quadratic polynomial satisfied by the period $\alpha=\zeta_5+\zeta_5^{-1}$ of the the $5th$ root of unity $\zeta_5$. Determine the ...
0
votes
0answers
30 views

Degree and basis of field extension $\mathbb{Q}[\sqrt{2+\sqrt{5}}]$

I want to find the degree and basis of the field extension $\mathbb{Q}(\sqrt{2+\sqrt{5}})$. let $\alpha=\sqrt{2+\sqrt{5}}$. $$\alpha^2=2+\sqrt{5},\quad \alpha^4-4\alpha^2-1=0.$$ So possible minimal ...
0
votes
0answers
27 views

Show that $f(x)$ factors in $K[x]$(whether or not $K$ is contained in the Galois closure $L$ of $f(x)$).

Let $f(x)\in F[x]$ be an irreducible polynomial of degree $n$ over the field $F$. For any $K$ is any Galois extension of $F$, show that $f(x)$ factors in $K[x]$ see this (whether or not $K$ is ...
4
votes
1answer
51 views

$\phi: K \to K$ be an $F$-embedding with ${tr.}\;{deg}(K/F)$ finite then $\phi$ is surjective.

Let $K$ be an algebraically closed field and $F$ be a subfield of $K$ with ${tr.}\;{deg}(K/F)$ finite. If $\phi: K \to K$ be an $F$-embedding show that $\phi$ is surjective. I know the result when ${...
1
vote
2answers
86 views

Factor $x^8-x$ in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$

Factor $x^8-x$ in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$ Here what I get is $x^8-x=x(x^7-1)=x(x-1)(1+x+x^2+\cdots+x^6)$ now what next? Help in both the cases in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$ Edit: I ...
1
vote
2answers
62 views

Proving that $[\mathbb{Q}(\sqrt{\sqrt{p+q}+\sqrt{q}},\sqrt{\sqrt{p+q}-\sqrt{q}}):\mathbb{Q}]=8$.

Some days ago I posted a question in MSE in order to correct a solution to the problem of Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}]=8$. After posting this another ...
0
votes
1answer
40 views

$x^2-3$ is separable over $\mathbb Q$ but not separable over $F_2$

$x^2-3=(x-\sqrt{3})(x+\sqrt{3})$ over $\mathbb Q$, so that part makes sense. Now, when it says $x^2-3$ is a polynomial over $F_2$, I imagine it means all the coefficients are calculated mod $3$, so $...
3
votes
2answers
40 views

Is it true that $Gal(K/F)\cong S_{n_1}\times \cdots S_{n_k}$?

I was reading galois theory and galois group from Dummit Foote and while reading Galois groups of polynomial a sudden question came into my mind that if $f(x)$ is an irreducible separable polynomial ...
2
votes
2answers
67 views

Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
0
votes
0answers
23 views

Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
3
votes
4answers
94 views

Galois group that is isomorphic to $\mathbb{Z}_3?$

I am curious if we can construct arbitrary number fields whose Galois group over $\mathbb{Q}$ is $G,$ for an arbitrary group $G.$ However, I cannot do this for $\mathbb{Z}_3.$ I think that if the ...
2
votes
2answers
39 views

Find the elements of the extension field using primitive polynomial over $GF(4)$

Let $p(z) = z^2 + z + 2$ be a primitive polynomial. I want to construct the elements of the extensional field $GF(4^2)= GF(16).$ Since $p(z)$ is primitive polynomial , it should generate the ...
1
vote
1answer
35 views

Normal Subgroup of Galois Group

Let $L/K$ be a Galois extension, and let $R\subseteq L$ be a subring such that $\tau(R)=R$ for every $\tau\in\text{Gal}(L/K)$. Let $\alpha\in R$. How would I show that $H=\{\tau\in\text{Gal}(L/K):\...
6
votes
1answer
42 views

Verifying if a given polynomial is primitive polynomial

Given a polynomial: $f(x) = x^2 + 2x + 2$ over $GF(3)$. I want to know if i can use it to construct $GF(3^2)$. My approach: This equation satisfies first condition: A primitive polynomial is ...
0
votes
1answer
24 views

The minimal polynomial satisfied by the primitive generator

I am trying to do the following problem that appears in Dummit and Foote book. But I have no idea how to start the problem. Can anyone please give me a hint? Thank you. Section 14.5, Problem #1(Page#...
2
votes
0answers
36 views

When is $S^G [G] \cong S$ as a $S^G[G]$-module?

Let $S$ be a ring and take a finite group $G$ acting on $S$. Put $R = S^G$, the fixed subring. When is $S$ a free rank-$1$ $R[G]$-module? That is, when do we have $S \cong R[G]$ as $R[G]$-modules? I ...
2
votes
2answers
54 views

cyclic extension of prime power of a local field

Let $K$ be a non archimedian local field of characteristic $p>0$ with residue field $\mathbb{F}_p$ and $l\neq p$ be a prime. It is known by local classfieldtheory that any abelian Galois ...
-2
votes
1answer
24 views

Existence of intermediate field [closed]

Let $K/F$ be a Galois extensio of degree $90$. To show that there is an intermediate field $L$ with $[L:F]=10$, do we have to check if there is a non trivial subgroup of order $10$ of a group of order ...
5
votes
2answers
149 views

Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
4
votes
2answers
63 views

An extension $K/F$ is Galois if for every simple extension $F \subset F(u) \subset K$, $[F(u):F] \leq 2$.

Let $F$ be a field, $char(F)\neq 2$ and let $K$ be an extension field of $F$. If for each $u\in K$, $[F(u):F]\leq 2$, show that $K$ is a Galois extension of $F$.
2
votes
0answers
31 views

Is the field of complex numbers the field of rational functions over a subfield?

My question is the same as the title. In fact, it is another question which leads me to think about this question: For which positive integer $n$ for which there is a subfield $K$ of $\mathbb{C}$ such ...
0
votes
0answers
38 views

Clarification on size of Galois group

I was under the impression that if $K/k$ is Galois then $G$ the Galois group of this extension had the size of the degree of the extension. However, I don't think this is correct anymore. Under what ...
1
vote
0answers
31 views

Torsion of elliptic curves and abelian extensions

Let $L/K$ be an abelian $p$-extension of number fields and $E$ be an elliptic curve over $\Bbb Q$. If $E[p](K)=0$, does it follow that $E[p](L)=0$ ? The converse is obviously true, but I don't have ...
5
votes
1answer
70 views

Equation with Galois group twisted $S_{3}$

I note that a Galois group is not just a Galois group. Let $r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$, denote the roots of a quartic equation. Then $x^4-5x^2+6$ has Galois group $Z_{2}^2$, where the ...
1
vote
0answers
26 views

Integrally Closed in Terms of Galois Theory

Let $K$ be a finite Galois extension of $\mathbb{Q}$ and let $A$ be a subring. We say $A$ is integrally closed in $K$ if any element $a \in K$ integral over $A$ is contained in $A$. Does being ...
0
votes
0answers
54 views

Separability in a finite field

Let $F$ be a finite field. Then any $f \in F[x]$ is separable. How do you tackle this? I thought of letting $f \in F[x]$ be irreducible, and let $\alpha$ be a root of $f$. Towards a contradiction, ...
2
votes
3answers
170 views

Characteristic and primitive roots of unity

Let $F$ be a finite field. If the characteristic of $F$ doesn’t divide $n$, then $F$ contains a primitive $n^{th}$ root of unity. I believe the converse is true, too, but I can’t prove either ...
4
votes
1answer
54 views

Invertible elements of $\mathbb{Z}_3[x] / (x^4+x^3-1)^3$

Let $F=\mathbb{Z}/3\mathbb{Z}$, $h(x)=x^4+x^3-1$, $R = F[x]/(h(x)^3)$. I know $R$ has $4$ ideals and $1$ maximal ideal. Let $M$ be the maximal ideal $(h(x))/(h(x)^3)$ I need to find the number of ...
1
vote
0answers
21 views

$G \neq gal(E:E_G)$?

Let $E$ be a field and $G$ be a set of automorphism of this field and $E_G$ be the field fixed by these automorphism. I can see that $G \subset gal(E:E_G)$. I was looking for an example to see when ...
2
votes
3answers
38 views

Can we have two different polynomials of the same degree $d$ here in the factorisation of $x^{p^n} -x$?

In the proposition "The polynomial $x^{p^n} -x$ is precisely the product of all the distinct irreducible polynomials in $\Bbb F_p[x]$ of degree $d$ where $d$ runs through all divisors of $n$." Can we ...
-1
votes
0answers
13 views

When is the norm of a field extension non-degenerate?

Let $\mathbb{E}\supseteq\mathbb{F}$ be a field extension. For each element $\alpha\in\mathbb{E}$ let $N(\alpha)\in\mathbb{F}$ be the determinant of the $\mathbb{F}$-linear function $\mathbb{E}\to\...
0
votes
1answer
49 views

f irreducible polynomial with $p-2$ real roots $\Rightarrow$ $Gal(\mathbb{Q}_{f}/\mathbb{Q}) \cong S_{p}$

I have no idea what to do to show the following Let $p\ge 5$ be a prime number 1) Let $H\subset S_{p}$ be a subgroup of the symmetric group. Assume that $p$ divides the order of $H$ and that $H$ ...
2
votes
1answer
70 views

$f(X):= X^n- x\in K[X]$ is irreducible and $Gal(K(y)/K)\cong \mathbb{Z}/n\mathbb{Z}$ if $y$ is a root of $f$

Hello I need help with solving the following task Let $K$ be a field with algebraic closure $\overline{K}$ and $K^{\times}:=K\backslash \{0\}$. Let $2\le n \in \mathbb{Z}$ such that $K^{\times}$ ...
0
votes
1answer
13 views

Any Sub-extension of a Radical Extension is Solvable?

I have seen the following theorem stated without proof. (I assume that all fields are characteristic zero.) Theorem: Let $F\subset K$ be a radical extension of fields. That is, suppose that $K$ can ...
1
vote
0answers
49 views

Does there exist an automorphism of $\mathbb{C}$ that does not keep $\mathbb{R}$ fixed? [duplicate]

I'm going to take Galois theory in the upcoming semester and I'm working on some basic problems right now. I was trying to figure out what $\mathrm{Aut}(\mathbb{C})$ should be. If $f: \mathbb{C} \to \...
0
votes
0answers
24 views

Noether Equation on Artin and Milgram Galois Theory book

On Section K Chapter 2 of Galois Book by Artin and Milgram it discusses Noether Equation, but I'm not sure what Noether Equation this is, I know about Noether Equation for symmetry in physics but not ...
2
votes
1answer
50 views

Is this a counterexample?

Suppose $K $ is a field and $\overline K $ an algebraic closure. Let $f $ be a $K $-automorphism of $\overline K$, let $L$ be the subfield of $\overline K $ fixed by $f $. In this post : (link), they ...
1
vote
1answer
65 views

Irreducible factors of $x^8 - x$ in $Z/2Z[x]$

I must find the irreducible factors of $f(x) = x^8 - x$ in $Z/2Z[x]$ and that's what I did: $f(x) = x(x-1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)$ but of course the correct solution is: $f(x) = x(x-1)...
2
votes
2answers
67 views

Primitive polynomial of a Galois field

How can one check that a polynomial is primitive polynomial or not? I have following polynomial $f(x) = x^3 + x^2 + 1$ and i want to know if i can use it to generate $GF(2^3)$. The definition i ...