Tagged Questions
1
vote
0answers
26 views
Characterization of normal extensions
I was wondering if there is a characterization of all $\alpha$ algebraic over $\mathbb{Q}$ such that $\mathbb{Q}(\alpha)$ is a normal extension over $\mathbb{Q}$. Also, is there method to prove that ...
2
votes
3answers
40 views
Questions about $\mathrm{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})$
Consider the extension $\mathbb{Q} \subset\mathbb{Q} (\sqrt{2}, \sqrt{3})$. How many elements are there in $\text{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})?$ Describe all elements in ...
0
votes
3answers
36 views
automorphisms and field extension $E$ of $\mathbb{Q}$.
I want a hint. That is all I ask for. The question I am asked to prove is as follows:
Let $E$ be an extension field of $\mathbb{Q}$. Show that any automorphisn of $E$ acts as the identity on ...
2
votes
0answers
30 views
Non-isomorphic simple extensions of the same degree of a field of positive characteristic
Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic.
I thought of an example where they are ...
5
votes
2answers
45 views
Galois Extensions and $n^{\text{th}}$ Roots
I've been studying for my prelims lately, and this problem has me stuck:
(a) Let $K$ be a field with no abelian Galois extensions. Suppose that $n$ is a positive integer and either $char(K)=0$ or ...
5
votes
1answer
42 views
finding fixed field of automorphism
Let $F$ be a field and let $g:F(x) \to F(x)$ be the automorphism which maps $x$ to $x+1$.
I need to find the fixed field of this automorphism.
So far I know $g$ fixes $F$. I want to use Galois ...
1
vote
1answer
91 views
Calculating The Galois Group of the Splitting Field of $f=x^3-3$
If we let $f=x^3-3$ the let L be the splitting field for this polynomial I am trying to find $\Gamma(L:\mathbb{Q})$ and all intermediate field extensions.
Now as this is a splitting field and finite ...
2
votes
2answers
65 views
Show $\zeta_p \notin \mathbb{Q}(\zeta_p + \zeta_p^{-1})$
I asked a question here: [Writing a fixed field as a simple extension of $\mathbb{Q}$ ], but realised I couldn't justify why the given quadratic was irreducible.
Thus: Is there a way of showing ...
2
votes
1answer
80 views
A basic question on factorization
Is the following true? If not, can anyone add some reasonable assumptions to make it true?
Statement. Let $E/F$ be a field extension and let $\alpha \in E \setminus F$ be algebraic over $F$. If $f(x) ...
1
vote
0answers
34 views
A question regarding linear disjiontness and the degree of a field extension
Let $K / k$ and $L / k$ be field extensions with $K$ and $L$ linearly disjoint over $k$. Suppose that $K' / K$ is an extension of finite degree, and that all fields are characteristic zero.
Then is ...
0
votes
1answer
159 views
A question regarding the finiteness of the degree of a field extension
Let $x$ and $y$ be transcendental and $y$ be algebraic over $\mathbb{Q}(x)$. Let $F$ be an algebraic Galois extension of $\mathbb{Q}(x)$ of infinite degree and let $\mathbb{Q}^{al}$ be the algebraic ...
3
votes
2answers
65 views
3 questions on field extensions
I am trying to figure out some things regarding field extensions and some questions have arisen on the way.
Let $a$ be a positive integer which doesn't have a rational $nth$ root:
Is the splitting ...
4
votes
3answers
115 views
Quadratic subfield of cyclotomic field
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
3
votes
1answer
76 views
Why don't I end up with the same splitting field?
I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
2
votes
1answer
43 views
Splitting field for $x^n+a$
What is a splitting field $E$ for $f(x)=x^n+a$ over the field $K$ of characteristic zero?
If I put $g(x)=x^{2n}-a^2=(x^n-a)(x^n+a)$. The splitting field $F$ of $g(x)$ is $K(\sqrt[n]{a},\alpha )$ ...
3
votes
1answer
73 views
Galois group of irreducible quartic with real coefficients
Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
3
votes
1answer
76 views
Irreducible polynomial over $\mathbb Z/7\mathbb Z$
Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
1
vote
2answers
52 views
Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$
Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
0
votes
1answer
29 views
Smallest Galois extension
Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
2
votes
1answer
38 views
Showing that a field extension is Galois
Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
3
votes
1answer
44 views
Lagrange interpolation of Galois field functions
I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
1
vote
1answer
57 views
Proving the order of a Galois group is equal to the dimension of $F$ over its fixed field w
Suppose $F/K$ is a finite dimensional field extension and $G = Aut_KF$. Let $G'$ be the fixed field of $F/K$, i.e. the set of members of $F$ which are fixed by every element of $G$. Before the ...
0
votes
0answers
34 views
System of equations and Abel theorem
Consider this system of 3 equations to be solved in x,y and z:
$a x^m=(y+z)^n$
$by^m=(x+z)^n$
$cz^m=(x+y)^n$
The parameters $(a,b,c)$ and the unknown $(x,y,z)$ are all in $ℝ₊$. Also, m and n are ...
4
votes
4answers
192 views
Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
2
votes
2answers
77 views
No rational solutions of a system of equations
Please show that there does not exist $(a,b,c)\in\mathbb{Q}^3$ such that
\begin{matrix}
a^2b+2b^2c+2ac^2=0\\
a^2c+ab^2+2bc^2=0\\
a^3+2b^3+4c^3+12abc=3.
\end{matrix}
I'm able to show that this ...
5
votes
1answer
138 views
An exercise on cyclic extensions of Hungerford's book, Algebra.
I'm trying to show the following exercise of the book, it is in the section of cyclic extensions and says the following:
Let $\overline{\mathbb Q}$ be a fixed algebraic closure of $\mathbb Q$, let ...
2
votes
2answers
48 views
Special elements of fields extensions
I was wondering if there is a method to find all elements $w\in F(\alpha_1,\ldots,\alpha_n)$ such that $F(w)=F(\alpha_1,\ldots,\alpha_n)$, where $\alpha_1,\ldots,\alpha_n$ are algebraic over the field ...
3
votes
0answers
85 views
Galois group of $x^{4}+ax^{2}+b$
Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$.
I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
2
votes
1answer
85 views
Splitting field that isn't a Galois extension
I'm trying to find a counter-example to following statement: if $K$ is the splitting field of $g\in F[x],$ then the extension $K/F$ is Galois.
I know the statement is true if $g$ is separable, ...
4
votes
1answer
43 views
No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$
Let $f(x) \in \mathbb Q[x]$ be irreducible of degree 4. Let $\alpha$ be a root of $f(x)$. Let $E = \mathbb Q(\alpha)$ and let $K$ be the splitting field of $f(x)$ over $\mathbb Q$. Prove that there is ...
1
vote
1answer
56 views
Degree of the splitting field of $x^{p^2} -2$ over $\mathbb{Q}$, for prime p.
I've already shown that the degree of the splitting field of $x^p-2$ over $\mathbb{Q}$ is $p(p-1)$ as follows:
$x^p-2$ has roots $\sqrt[p]{2}\omega_{k}$ for $k=0,1,...,p-1$, where the $\omega_{k}$ ...
0
votes
0answers
62 views
Trace and cyclotomic field
Let $K=\mathbb{Q}(\zeta_p)$ be the cyclotomic field of $p$th roots of unity for the prime $p$ and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Let $\zeta$ denote any $p$th root of unity. Please show that ...
2
votes
3answers
52 views
Show that for $n \geq 2$, the $n^{th}$ cyclotomic polynomial is a reciprocal polynomial, i.e. $\Phi_{n}(x) = x^{\phi(n)}\Phi(n)(x^{-1})$.
Here $\phi(n)$ is the Euler totient function and the degree of $\Phi_{n}(x)$.
What I've done so far:
Let $\phi(n) = p$ so the following products each have p components.
$$\Phi_{n}(x) = \prod_{k=1, ...
0
votes
1answer
33 views
A radical extension with a non-radical subextension
For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
2
votes
1answer
40 views
Show the extension is not cyclic
Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
2
votes
3answers
56 views
Two quadratic fields over $\mathbb{Q}$
I'm having a bit of trouble showing that the two quadratic fields $\mathbb{Q}[X]/(X^2+1)$ and $\mathbb{Q}[X]/(X^2+3)$ over $\mathbb{Q}$ are not isomorphic (as fields). Could someone help me? Perhaps ...
1
vote
1answer
100 views
Fixed field of automorphisms determined by $t\mapsto at+b$.
Suppose $E=\mathbb{F}_p(t)$, the field of rational functions in a transcendental $t$ over the finite field of $p$ elements. Suppose $G$ is the group of field automorphisms fixing $\mathbb{F}_p$ ...
1
vote
1answer
49 views
Irreducibility over Fp - A useless hint?
this is my first question here so bear with me.
Dummit and Foote 13.5.5:
For any prime p and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$
The ...
4
votes
2answers
76 views
Radical extension
Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$.
It's ...
1
vote
1answer
75 views
Fixed field of group of automorphisms on $\mathbb{C}(t)$.
I've been stuck on this problem tonight.
Suppose $E=\mathbb{C}(t)$ where $t$ is transcendental over $\mathbb{C}$, and let $\omega$ be a primitive cube root of unity. Let $\sigma$ be the automorphism ...
3
votes
2answers
157 views
How to find the splitting field and Galois group of $x^6 -4x^3 +1$?
I am trying to find the splitting field $L$ of the $x^6 -4x^3 +1$ over $\mathbb{Q}$, and its Galois group.
Here are some things I have figured out. I did the usual trick of solving for $x^3 = 2\pm ...
0
votes
1answer
52 views
Can any monomorphism of a subfield of a splitting field be extended to an automorphism?
It's a common theorem in field theory that if $\varphi: F\to\overline{F}$ is a field isomorphism, and if $E$ and $\overline{E}$ are splitting fields of monic polynomials $f(x)$ and $\overline{f}(x)$, ...
4
votes
1answer
57 views
field of algebraic numbers
Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals?
Many Thanks
2
votes
1answer
53 views
Galois Theory Problem (Fundamental theorem of Galois)
Let $k$ be a field of characteristic$>2$. Let $c\in k$, $c\notin k^2$. Let $F=k(\sqrt{c})$. Let $\alpha=a+b\sqrt{c}$ with $a,b\in k$ and not both $a,b=0$. Ket $E=F(\sqrt{\alpha})$. Prove that the ...
3
votes
0answers
102 views
Question about solvability of the minimal polynomial of the element in extended field.
Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$.
Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
1
vote
1answer
80 views
splitting field of $x^p-a$ over $\mathbb{Q}$ has no primitive $p^2$ roots of unity
It is known that the splitting field of $x^p-a$ over $\mathbb{Q}$ has no $p^2$ roots of unity. We can assume $a\in \mathbb{Q}$ is not a pth power in $\mathbb{Q}$. I came up with the following proof of ...
6
votes
2answers
94 views
Compositum of fields with trivial intersection
Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
6
votes
1answer
122 views
How to compute the Galois group of $x^5+99x-1$ over $\mathbb{Q}$?
I was stuck trying to compute the Galois group of $x^5 + 99x -1$. The problem asks to compute the Galois group over $\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_5, \mathbb{F}_{11}$ and $\mathbb{Q}$. I was ...
0
votes
3answers
67 views
A sum of products symmetric in the images under all the embeddings
Let $\mathbb{Q}\subset K\subset \mathbb{C}$ where $K$ is a finite extension of $\mathbb{Q}$. Let $\sigma_1, \dots, \sigma_n$ be all the embeddings $K\rightarrow \mathbb{C}$. Is it true that elements ...
2
votes
1answer
88 views
Number of monomorphisms $\mathbb{Q} \to \mathbb{C}$
In my abstract algebra class, we have been tasked to find all monomorphisms $\mathbb{Q} \to \mathbb{C}$. The book (Stewart's Galois Theory) gives an example for $\mathbb{Q}(\sqrt[3]{2}) \to ...
