Questions tagged [galois-extensions]
For questions about Galois extensions fields. We say $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.
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Extension degree related to a transcendental extension in positive characteristic
Let $k$ be algebraically closed field of characteristic $p > 0$. Let $k(t)$ be a transcendental extension. Let $L$ be the splitting field of $x^n-t$ over $K$ $(n \geq 1)$. Let $G = Aut(L/K)$ and $F ...
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Is the following statement about splitting fields true?
Let us take a look at the following definitions:
Let $F$ be a field and $f(x)\in F[x]$ then a field extension $E$ of $F$ is said to be the splitting field of $f$ over $F$ if $f$ splits completely in $...
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Is $i$ an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$?
I am trying to check if $i$ is an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$. How can I check if this is the case? Would it be correct to express my field as $a\mathbb(\sqrt[4]{5}) + b ...
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Infinitely large Galois extensions of $\mathbb Q$ inside $\mathbb Q_p$
Let $p$ be a fixed prime number and denote $\mathbb Q_p$ the field of $p$-adic numbers.
For each positive integer $n$, I would like to construct a finite Galois extension $K/\mathbb Q$ of degree at ...
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Compositum of intersection fields
Suppose that $K_1$, $K_2$ and $K_3$ are finite Galois extensions of a field $k$. Let $(K_1 \cap K_3)(K_2 \cap K_3)$ be the compositum of $(K_1 \cap K_3)$ and $(K_2 \cap K_3)$. Is it always true that $$...
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Degrees of field extensions
Find the degrees of $\Bbb{Q(a)/Q, Q(a,b)/Q, Q(a,b,c)/Q}$ where $a,b,c$ are the roots of $x^3+x-1$ where a is real.
To solve this problem, I noticed that $x^3+x-1$ is irreducible. Then why isn’t the ...
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Tensor product of finite Galois extensions
Let $K_1$ and $K_2$ be finite Galois extensions of $k$, set $G_k = \text{Gal}(K_1 K_2/k)$, and $H= \text{Gal}(K_1 K_2/K_1 \cap K_2)$.
I want to prove that as $k$ algebras,
$$K_1 \otimes_k K_2 \cong \...
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Find a particular intermediate field $M$ such that $\mathbb{Q}\subset M\subset\mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})$.
Problem: Let $\alpha=\sqrt{\frac{3+i\sqrt{7}}{2}}$ and $K=\mathbb{Q}(\alpha)$. Find the fixed field $M=\{x\in \mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})|\sigma(x)=x\}$, where $\sigma$ is the ...
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Connection between derivatives in Galois theory and calculus
In field theory, we can define a function
$d:F[X]\to F[X]$ where $\sum_{i=0}^n a_iX^i \mapsto \sum_{i=1}^n ia_iX^{i-1}$, where $F[X]$ is some polynomial ring.
Letting $d(f)=f'$ for $f\in F[X]$, I have ...
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$K(u, v) = K(u+v)$ when characteristic of $K=0$?
Let $F$ be a field extension of $K$ where $u, v$ are in $F$ such that $u$ is separable over $K$ and $v$ is purely inseparable over $K$. The answer to this question shows that $K(u, v) = K(u+v)$ when ...
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Finding a quartic polynomial whose resolvent cubic is given
Recently I was assigned a homework problem to determine the possible Galois groups of an irreducible quartic polynomial over $\mathbb{Q}$ that has exactly two real roots (equivalently has negative ...
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Existence of $p^{th}$ root in an inseparable extension
Let $K= \mathbb{F}_p(t)$ i.e. the field of rational functions over $\mathbb{F}_p$.
I'm trying to prove that if $x$ is inseparable over $K$, then $K(x)$ contains a $p^{th}$ root of $t$.
I tried to find ...
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A dimension problem related to an abelian extension of a field
Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-vector space $V$ to be $F$-span ...
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Show that $\mathbb{Q}(\sqrt{2+\sqrt{2}}): \mathbb{Q}$ is normal.
My Attempt: Let $\gamma_0=\sqrt{2+\sqrt{2}}, \gamma_1=-\sqrt{2+\sqrt{2}}, \gamma_2=\sqrt{2-\sqrt{2}}, \gamma_3=-\sqrt{2-\sqrt{2}}$. Let $L=\mathbb{Q}(\gamma_0, \gamma_1, \gamma_2, \gamma_3)$ and $M=\...
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Isomorphism of Galois Groups
$E$ is the splitting field for $f(x)=x^3-2$ and $K := E(\sqrt 5)$. We want to show that $G=\text{Gal}(K/\mathbb{Q}) \cong \mathbb{Z_2} \times S_3$.
To do this, we know that $K$ has subfields $E$ and $...
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A polynomial is irreducible in a splitting field of another polynomial
Suppose that $E$ is a splitting field over $\mathbb{Q}$ for $f(x)=x^3-2$. We want to show using Galois theory that $g(x)=x^2-5$ is irreducible in $E[x]$.
Here is what I have so far. Assume for ...
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What is the Galois group of $x^n + 2x^{n-1} + \dots + nx - 1$?
I computed the Galois group of $$P_n := x^n + 2x^{n-1} + \dots + nx - 1$$ for $n=2,3,4,5,7$ with the help of Dedekind's theorem, and each time it was $S_n$. So it made me wonder: is it always $S_n$? ...
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Testing membership of elliptic curve points in subgroups of binary extension fields
I come from an engineering background. Let a binary extension field $GF(2^{233})$ and a finite group $E$ made of points defined by a curve over Binary Fields i.e. $y^2 + xy = x^3 + x^2 + b$ with ...
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Chapter 1, Proposition 14 of Lang's number theory - further explanation for normality
Attached is prop 14 on page 15 of Lang's 'algebraic number theory' 2ed. Note, $G_\mathfrak{B} \subset G$ is the subgroup fixing $\mathfrak{B}$
Can someone perhaps add a sentence or two to explain ...
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Galois group of a function field over finite field
I have a question about the structure of this Galois group that I can't understand:
suppose that $p>2$ is prime and $q$ is any power of p, and we have these two function fields:
$$K=\mathbb{F}_{p}(...
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On the action of Galois group on some other group
Let $L/K$ be a cyclic Galois extension of degree $5$ of number fields, such that $Gal(L/K) = \langle \sigma \rangle$. We consider an action of $Gal(L/K)$ on some groupe $G$. The action is defined as: :...
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Commutative ring Galois extension with two non-isomorphic Galois groups
I'm currently studying about the generalisation of Galois Theory for commutative rings. I'm trying to find a good example that illustrates how this generalisations allows a Galois extension of a ...
- 3,420
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Why F is galois extension of the fixed field of Galois group of F over K, where, F is the extension field of the field K
I am able to show that Galois group of F over K is same as Galois group of F over the fixed field of the same galois group. I think since both the groups are equal so their fixed fields are the same ...
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Enumerate the $\mathbb{Q}$-automorphisms of the normal extension $\mathbb{Q}(\sqrt[p]{2},\omega_p)$
Calling $\omega_p$ a primitive p-th root of unity, Wikipedia states that:
$\mathbb{Q}(\sqrt[p]{2},\omega_p)$ is a normal extension of $\mathbb{Q}$
This normal extension has order $p(p-1)$
Also, I ...
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Discovery of the relationship of the Theorem 18.5 in Ash & Gross's "Fearless Symmetry" book
In Ash & Gross book: Fearless Symmetry, Chapter 18, the following Theorem is stated:
THEOREM 18.5: Let q be a prime other than p that is
unramified for the Galois representation ψ. (This will be ...
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$K := \mathbb{Q}( \sqrt{5}, \sqrt{−1})$ is a Galois extension of $\mathbb{Q}$ of degree 4 [duplicate]
we just had a a few pages about Galois theory and I have trouble understanding it. I want to prove that $K := \mathbb{Q}(
\sqrt{5},
\sqrt{−1})$ is a Galois extension of $\mathbb{Q}$ of degree 4 and ...
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How to see that the cyclotomic extension is a separable extension?
$\mathbb{F} \subset \mathbb{E}$ is a cyclotomic extension, if
$\mathbb{E}$ is the splitting field of $x^n-1$ over $\mathbb{F}$.
We were directly told that $\mathbb{E} \mid \mathbb{F}$ is a Galois ...
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Galois group of $x^4-11$
The splitting field of $x^4-11$ over $\mathbb{Q}$ is $\mathbb{Q}(i, \sqrt[4]{11})$, so $\mathbb{Q}(i, \sqrt[4]{11})/\mathbb{Q}$ is a Galois extension. I managed to prove that the Galois group of this ...
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The automorphism group of $E(\alpha)/F(\alpha)$
Suppose $E/F$ is a Galois extension. Let $\alpha$ be an (for simplicity, suppose $\alpha$ is algebraic over $F$) element of some extension $K$ of $E$. I have two questions:
Is $E(\alpha)/F(\alpha)$ ...
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Intersection of fixed fields
In my Galois theory course we are currently finding fixed fields. In an exercise we defined the following isomorphisms
$$\sigma(\alpha) = \alpha i,\quad \sigma (i) = i, \quad o(\alpha)=4$$
$$\tau(\...
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Prove that if $N, M \leq Aut (K/F) $ in a galois extension $K/F $, the fixed field of $\langle N, M \rangle $ is $F^N \cap F^M $
The Fundamental Theorem of Galois Theory states that in a Galois extension $K/F $, there is a bijection of subfields $E $ of $K$ containing $F $ and the subgroupsn $H $ of $G$. For any subgroup $N \...
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Extensions of finite fields and normality.
I have two question about field theory and field extension more precisely. All i have to do is say if these are true or false, give a demonstration if true and a coutner-example if false.
If $\mathbb{...
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Galois Action on Absolutely Irreducible Polynomials
Consider an absolutely irreducible polynomial $f(x)$ with coefficients in $\mathbb{F}_{q^r}$. Consider also $Gal(\mathbb{F}_{q^r}, \mathbb{F}_q)$. Is there any possibility that $\sigma(f(x))$ remains ...
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Galois Correspondence as it relates to $\mathbb{Q}(\sqrt{1 + \sqrt{2}})$
Tl;dr
Before getting into the details, let me just give a summary of the main source of my confusion. Basically, I currently believe the following list of statements are true:
The extension $\mathbb{...
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Galois: Why does $\{1,f,f^2\}$ correspond to $\mathbb{Q}(\omega)$?
I'd like to discuss this example from the wikipedia article.
It claims that the group $\{1,f,f^2\}$ corresponds to $\mathbb{Q}(\omega)$. But then it must hold true that the fixed field $Fix(\{1,f,f^2\}...
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Countability of Gal(C/Q) and isomorphism of its subgroups [closed]
Firstly, I want to know how can I prove that Gal(C/Q) is an uncountable group. Secondly how to show that the subgroup {g ϵ Gal(C/Q)| g continous} is isomorphic to Z/2. Really I don't know how to start ...
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About the primitive elements of $\mathbb{Q}(\sqrt{3},i\sqrt{5})$
(I have already prove that $\mathbb{Q}(\sqrt{3},i\sqrt{5})=\mathbb{Q}(\sqrt{3}+i\sqrt{5})$ in case is useful); now I am asked to prove that if $v\in \mathbb{Q}(\sqrt{3},i\sqrt{5})$ has the property ...
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What is the Galois group of this extension?
Let $u$ be the real root of $x^3-x+1$; I am trying to calculate de Galois group of $\mathbb{Z}/3\mathbb{Z}(u)/\mathbb{Z}/3\mathbb{Z}$; let's start with the first one; as $x^3-x+1$ is monic, and ...
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Is true that the number of elements in Gal$(F(u)/F)$ is equal the number of distincts roots of its minimal polynomial on $F(u)$?
Suppose we have an extension field $K$ over $F$ and $u$ an algebraic element of $K$. If $p(x)$ is its minimal polynomial in $F[x]$, is true that the order of Gal$(F(u)/F)$ equals the number of ...
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If $|Gal(K/F)|=[K:F]$ then $K/F $ is a Galois extension
I'm using the following definition: Let $K$ be a finite-dimensional extension field of $F$. Then $K$ is Galois over $F$ if the fixed field of Gal$(K/F)$ is $F$.
Suppose now that we have a extension ...
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Skew fields with nonzero characteristic
There are many ways to construct a skew field of nonzero characteristic, e.g. the universal field of fractions of a skew polynomial ring $E[x;\sigma]$, a suitable choice of a quaternion algebra over $...
- 1,507
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Simple argument that field automorphisms can extend
I've been learning about Galois theory and in a lot of proofs, an automorphism $\sigma\in Gal(L/K)$ is extended to $\hat{\sigma}\in Gal(M/K)$, when $K\subseteq L\subseteq M$ and $M/K$ and $L/K$ are ...
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why is $\sigma\tau\neq\tau\sigma$ in $\operatorname{Gal}(\mathbb{Q}(\sqrt[3] 2,\zeta_3)/\mathbb{Q})?$
Theorem: $\operatorname{Gal}(\mathbb{Q}(\sqrt[3] 2,\zeta_3)/\mathbb{Q}) =\langle\sigma, \tau\rangle\cong S_3$
My question: Why is $\sigma\tau\neq\tau\sigma$ in $\operatorname{Gal}(\mathbb{Q}(\sqrt[3]...
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Galois extension of $p^2$ root of unity contains a subfield whose Galois group is $\mathbb{Z}/p\mathbb{Z}$.
Let $\zeta$ be a primitive $p^2$ th root of unity with prime $p$. Let $L$ be the Galois extension $\mathbb{Q}(\zeta)$,
I want to prove $L$ contains a subfield whose Galois group is $\mathbb{Z}/p\...
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When are the extensions $L(S)/K$ and $L(S)/L$ totally ramified?
Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$. Let us consider the polynomial ring $R=K[x_1,x_2,...,x_n]$ in $n$-variables and $f_1, f_2, \cdots, f_m \in K[x_1, \cdots, x_n]$.
...
- 9,500
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Linearly disjoint family of fields $\{L_i \}_{i=1}^n$ over $K$
Suppose that $K/\mathbb{Q}$ is a number field.
$(*)$ Let $L_1,\dots,L_n/K$ be a finite family of finite Galois extensions such that $L_i\cap L_j = K$ for all $i,j$.
Is this enough to conclude that $...
- 1,137
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2
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Not surjective function by using algebra tools
Let $T$ be a transcendental basis and $E\subset T$ over $\Bbb Q$ and let $f\colon \Bbb R\to \Bbb Q\cdot E$ such that $f(0)=0.$ Put, $D:=\{x\colon f(x)\neq 0\}.$ Define a function $g\colon \Bbb R\to\...
- 2,153
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Two elements are not equal by using algebraic independent trick
Definition. $\bar{\Bbb Q}(S)$ denotes an algebraic closure of ${\Bbb Q}(S)$ in $\Bbb R$, that is, $\bar{\Bbb Q}(S)$ is the set of $x\in\Bbb R$ that are algebraic over $\Bbb Q(S).$
It seems that I ...
- 2,153
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75
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Computing a certain Galois group
I am trying to answer this question: Suppose $p$ is an odd prime and $K/\mathbb{Q}$ is the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of 1 in $\mathbb{C}$. (a) Show that $K$...
- 521
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How can you specifically extend an automorphism from a quadratic field to one of a cyclotomic field?
I am trying to see how can I extend the automorphisms of the Galois extension $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$, for $d$ square-free, to automorphisms of $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ that fix $\...
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