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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2
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3answers
36 views

Why number of bases of $\mathbb{F}_p^2$ equals order of $GL_2(\mathbb{F}_p)$?

Artin, Algebra, Chapter 3, Ex. 4.4 I can prove (b), viz., that The order of $GL_2(\mathbb{F}_p)=p(p+1)(p-1)^2$ The order of $SL_2(\mathbb{F}_p)=p(p+1)(p-1)$ However, I have no idea how to prove (a)...
0
votes
0answers
19 views

Field of the form $\{a+bi|a,b\in \mathbb{F}_p\}$

Artin, Algebra, Chapter 3, Ex 1.11 Consider whether the set of symbols $\{a+bi|a,b\in\mathbb{F}_p\}$ forms a field, if the laws of composition are made to mimic addition and multiplication of complex ...
1
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2answers
22 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
1
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0answers
43 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
8
votes
2answers
452 views

What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
1
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0answers
27 views

Minimal polynomial of a primitive element for Galois extensions with Galois group $S_n$

Let $K$ be a global field, $f(x)\in K[x]$ be an irreducible separable polynomial and $L$ be the splitting field of $f(x)$. Suppose that the Galois group of $L$ over $K$ is the symmetric group $S_{\deg(...
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1answer
54 views

Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[...
4
votes
1answer
36 views

Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, \sqrt{f}]$....
2
votes
1answer
50 views

About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n $ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n $ ( A subgroup $G$ ...
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3answers
46 views

rational numbers field axioms

Let $\mathbb Q$ be the rational number field. Is the group $K=\left\{\left.\begin{pmatrix} a & 2b\\ b & a \end{pmatrix}~\right|~ a,b\in \mathbb Q\right\}$ a field with the regular addition ...
1
vote
3answers
26 views

Finding the fixed subfield (Galois theory)

Let's say we are working with the field extension $\mathbb{Q}(\gamma)$, where $\gamma$ is the seventh root of unity. I know my basis for this extension will thus be: $\{1, \gamma, \gamma^2, \gamma^...
0
votes
1answer
41 views

By “shifting” , what does this mean?

I am looking at the solutions to a problem that asks me to show The only subfields of $\mathbb{Q}(i,\sqrt{5})$ are $\mathbb{Q},\mathbb{Q}(i),\mathbb{Q}(\sqrt{5}),\mathbb{Q}(i\sqrt{5}),\mathbb{Q}(...
0
votes
1answer
40 views

Some natural question on subfield of Galois extension

Let $\alpha,\beta\in \mathbb{\overline{Q}}$ and assume $\deg(\text{Irr}(\alpha,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q}))=\deg(\text{Irr}(\beta,\mathbb{Q(\alpha)})=2$. Then I strongly guess that ...
0
votes
1answer
43 views

If $f(a)=f(a+1)$, then $F$ has characteristic $0$.

Suppose $f\in F[x]$ is irreducible, $E$ is the splitting field of $f$, and for some $a\in E$ we have $f(a)=f(a+1)=0$. Then $F$ has characteristic $0$. I'm not sure how to use the last assumption: If ...
0
votes
1answer
14 views

Finding the fixed subfield corresponding to a cyclic subgroup of the Galois group

Let's say I have a field extension $E$ of some field $F$ and I also know the Galois group of $E$ over $F$. Suppose I have a subset of this Galois group which is cyclic, thus generated by some ...
1
vote
1answer
49 views

Degree of the field extension

I need to determine the degree of the field extension $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3}}))/\mathbb{Q}$. I've determined that the minimal polynomial of $\sqrt{(2+\sqrt{2})(3+\sqrt{3}})$ is $$f(...
1
vote
1answer
24 views

Prove that $F \subset \sigma L$ is also radical.

Suppose that we have finite extensions $F \subset L \subset M$ and $\sigma \in Gal(M/F)$ and assume that $F \subset L$ is radical. Prove that $F \subset \sigma L$ is also radical. Since the ...
7
votes
1answer
50 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...
1
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1answer
39 views

The elements and operations of the field $C = \Bbb R[x] / \langle x^2 + 1 \rangle$

$$C = \Bbb R[x] / \langle x^2 + 1 \rangle = \{[a + b x_{x^2 + 1}]\}$$ I know $C$ is a field since it has complex roots $(x+i)(x-i)$ and is irreducible over the reals, also since deg is $2$. How ...
3
votes
1answer
66 views

When does a f.g. algebra over a field $F$ make it “look like $F$ is algebraically closed?”

Let $F$ be a field, and let $A$ be a finitely generated algebra over $F$. If $\mathfrak m$ is a maximal ideal of $A$, then $A/\mathfrak m$ is an algebraic extension of $F$, although it is in general ...
1
vote
1answer
38 views

All intermediate sub extensions of $\mathbb{Q} \subseteq K \subseteq \mathbb{Q}(\zeta_8)$.

I know there is a similar question posted on Stack Exchange, however it deals with periods, and I do not understand the solutions provided. I know that the Galois Group of the field extension $\...
2
votes
1answer
42 views

Extension of field homomorphisms and pullback square

Let $E/k$ and $F/k$ be two subextension of a field extension $K/k$. The following square induced by restriction functions is always pullback square (in category of sets and functions)? $$\begin{...
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votes
2answers
48 views

Intersection of two subfields of $F(X)$ [duplicate]

Let $E=F(x)$ for a field $F$ of characteristic $0$. Show that $F(x^2) \cap F(x^2-x) = F$ as subfields of $F(x)$. I could use a hand with this... Thanks
2
votes
3answers
52 views

prove that if $\alpha \in \mathbb{Q}(\sqrt{2},\sqrt[4]{2},\sqrt[8]{2},…)$ then $\alpha \in \mathbb{Q}(\sqrt[2^n]{2})$ for some $n$

prove that if $\alpha \in \mathbb{Q}(\sqrt{2},\sqrt[4]{2},\sqrt[8]{2},...)$ then $\alpha \in \mathbb{Q}(\sqrt[2^n]{2})$ for some $n$. I have the solutions which state: Since $\alpha \in \mathbb{Q}(...
1
vote
1answer
28 views

Galois extension of intersection of fields

I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks
3
votes
1answer
30 views

About some properties of composites of field extesions

When I'm self-studying Parick Morandi's book Field and Galois Theory,I came across some problems,which I can't work out fully. Let $K$, $L$, be two extension fields of base field $F$. If $K/F$ and $...
1
vote
2answers
48 views

Show that $x^{n-1} =1$ for all non-zero element in a field

Question: Show that $x^{n-1} =1$ for all non-zero element in a field Let F be a finite field of order n. Show that $x^{n-1}=1$ for all non-zero $x \in F$. We have $\left | F \right |=n.$ ...
8
votes
1answer
125 views

Can $\cos (2\pi/7)$ be written as $p+\sqrt{q}+\sqrt[3]{r}, p,q,r\in \mathbb{Q}$?

Is it possible to find $p,q,r \in \mathbb{Q}$ such that $$\cos \frac{2\pi}{7}=p+\sqrt{q}+\sqrt[3]{r}.$$ Assume we can find such $p,q,r$, then $\mathbb{Q}(\cos \frac{2\pi}{7}) \subseteq \mathbb{Q}(\...
0
votes
1answer
23 views

Show that Gal$(E/\mathbb{Q})$ is abelian, where $E$ is the splitting field of $f(x)=x^{14} - 1$

Let $E$ denote the splitting field of $f(x)=x^{14}-1$. I want to show that the Galois group is abelian. Here's my attempt: The different 14'th roots of unity are given by $w=e^{i \pi n/7}$ where $n = ...
0
votes
1answer
20 views

Give all extensions of the mapping to an isomorphic mapping

Let $ E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}). $ It can be shown that $ [E : \mathbb{Q}] = 8. $ For each given isomorphic mapping of a subfield of $ E, $ give all extensions of the mapping to an ...
0
votes
2answers
26 views

Prove that $ \sigma(x) $ and $ x $ are conjugate over $ F $

Let $ E $ be an algebraic extension $ F $ and $ x \in E $ and $ \sigma: E \to E $ be an automorphism of $ E $ fixing $ F. $ Prove that $ \sigma(x) $ and $ x $ are conjugate over $ F. $ I am starting ...
4
votes
2answers
74 views

Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
1
vote
1answer
48 views

Why is the Galois group for a cyclotomic extension isomorphic to $Z_n^{\times}$ and not to $Z_{n-1}$?

I'm struggling to understand this. Let $\mathbb{Q}(\xi)$ be the splitting field for $x^n - 1$, so here $\xi$ is the primitive n-th root of unity. If I consider the possible automorphisms of this ...
0
votes
0answers
23 views

Is $\prod_{\sigma\in \Sigma(L)}\mathbb Z$ free of rank $[K:\mathbb Q]$?

Let $L/\mathbb K$ be a Galois extension of number fields and $\Sigma(L)$ the set of embeddings of $L$ into $\mathbb C$. Let $H_L= \prod_{\sigma\in \Sigma(L)}\mathbb Z$. Let $G$ act on $H_L$ in the ...
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vote
0answers
16 views

Proof of $\Bbb R$ is the unique complete linear order.

I'm looking for the theorem that says that all linearly ordered, complete fields are isomorphic. I couldn't find references online, but I'm sure this theorem must have some name. A link would be ...
1
vote
1answer
42 views

constructing a Galois group for a cyclotomic extension of $\mathbb{Q}$

Suppose that our polynomial is $x^5-1$, thus the splitting field is $\mathbb{Q}(\gamma)$ where $\gamma$ is a primitive 5-th foot of unity. Then our basis for the extension field will be: $\{1, \...
1
vote
1answer
27 views

Showing that a field $k$ is a splitting field for $p(x) \in \mathbb{Q}$

Suppose $\gamma$ is the fifth root root of unity. That is, $\gamma = e^{\frac{2\pi i}{5}}$, so $\gamma$ is a root of $p(x) = x^5-1$, or more precisely of $x^4+x^3+x^2+x+1$ since we can factor out a $(...
2
votes
0answers
52 views

Embedding a number field in $\mathbb{Q}_p$.

Let $K/\mathbb{Q}$ be a finitely generated field extension and let $(x_1,\cdots,x_m)$ be a transcendance basis of $K/\mathbb{Q}$. Using primitive element theorem, there exists $y\in K$ algebraic over $...
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votes
0answers
17 views

Calculating the fixed subfield of a splitting field $E$ corresponding to a subgroup of the Galois group $G = G(E/\mathbb{Q}$)

Here my splitting field is $E = \mathbb{Q}(\sqrt[3]{3}, \gamma)$, where $\gamma$ is a primitive cube-root of unity. This is the splitting field for $x^3-3$ in $\mathbb{Q}[x]$. I have calculated ...
0
votes
0answers
19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
4
votes
1answer
65 views

Abelian Galois group of $f$ implies splitting is simple extensions by a root of $f$.

Given an irreducible polynomial $f\in \mathbb{Q}[x]$ with Abelian Galois group, I would like to show that the splitting field $K$ over the rationals can be written as a simple extension $\mathbb{Q}(\...
1
vote
0answers
33 views

Is this a Galois extension?

I have the two simple extensions $F \subseteq F(\theta)$ and $F \subseteq F(\gamma)$, which are stated to be Galois extensions. We also have char$(F) = 0$. The problem is whether or not $F \subseteq F(...
2
votes
1answer
36 views

For what $a,b \in \mathbb Z$ is $\frac12(\sqrt a+\sqrt b)$ an algebraic integer

I've recently been working on a practice midterm for my number theory class, and here is a problem I've come across. As there are no solutions posted, I'd like to verify that what I'm doing is ...
1
vote
3answers
50 views

Is $\mathbb{F}_{81}$ a field extension of $\mathbb{F}_{27}$?

Is $\mathbb{F}_{81}$ a field extension of $\mathbb{F}_{27}$? If it is, what is $[\mathbb{F}_{81}:\mathbb{F}_{27}]$? In this case, $\mathbb{F}_{81}$ means a field with 81 elements. I know like $\mathbb{...
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0answers
38 views

$k(x)$ is rational function field over $k$. How to find an element $u(x)$ such that $k(u)=k(x+\frac{1}{x})\bigcap k(x-x^2)$? [closed]

Let $k$ be a field, and $k(x)$ be the rational function field in one variable over $k$. $L_{1}=k(x+\frac{1}{x})$ and $L_{2}=k(x-x^2)$are two intermediate fields of the extension $k(x)/k.$ Find an ...
2
votes
3answers
70 views

Finding a minimal polynomial of an algebraic element using Galois theory

There is a canonical (but difficult) way of determining the minimal polynomial of an algebraic element $\alpha$ in a field $F$, namely by considering the $F$-linear transformation defined by left ...
0
votes
0answers
37 views

Is this extension of the real numbers a field? It involves a unit of infinity.

Is this extension a field? Or perhaps some other structure? The extension depends on two basic ideas: A definition for a unit of infinity, the same as one given by Roger Penrose, and The infinity ...
0
votes
1answer
25 views

Why $S\cong A/I$?

Let $A$ a $\mathbb K-$algebra of finite dimension where $\mathbb K$ is an algebraically closed field and let $S$ a simple $A-$module. In the proof of the Schur lemma, it says that since $S\cong A/I$ ...
0
votes
1answer
64 views

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$.

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$. My try: Any finite field $F$ is of the order $p^n$ where $p$ is a prime and $n\in \Bbb N$ . If we want to make a field of ...
1
vote
1answer
61 views

Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.

In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in F[x]$,...