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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Finding a condition

So, we have $Q(\sqrt{2},\sqrt{3})=Q(\sqrt{2}+\sqrt{3})$. $\supset$ is absolutely trivial and $\subset$ holds because $\frac{1}{\sqrt{3}+\sqrt{2}} = \sqrt{3}-\sqrt{2} \in Q(\sqrt{2},\sqrt{3})$ and ...
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1answer
57 views

Field theory: an equality involving the number of homomorphisms from an extension $E$ of $F$ to $\overline{F}$

First some notation. Let $F$ be a field, $E$ an algebraic extension of $F$ and $\overline{F}$ the algebraic closure of $F$. Let $\{E:F\}$ represents the number of non-zero homomorphisms from $E$ to $...
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2answers
50 views

Finding $p(x)$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$

I am trying to find a polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$. This is what I tried to do: Consider ...
5
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1answer
50 views

Showing normalizer of Galois group

Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove ...
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2answers
43 views

Showing automorphisms on $\mathbb{C}(x)$

Let $\mathbb{C}(x)$ denote the field of rational functions over $\mathbb{C}$, the field of complex numbers. Consider the six mappings $\phi : \mathbb{C}(x) → \mathbb{C}(x)$ defined by $\phi_{1}:f(x) ...
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2answers
54 views

Why is $\mathbb{Q}(\sqrt{2}\sqrt[3]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$

Why is $\mathbb{Q}(\sqrt{2}\sqrt[2]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$ "obvious"? My book states this as obvious, but then proves the opposite inclusion. I would have thought that $\mathbb{...
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1answer
33 views

Field extension if element fixed by only identity [closed]

Suppose that $E$ is a Galois extension of $F$ and that $α \in E$ is left fixed by only the identity in $\text{Gal}(E/F)$. Prove that $E = F (α)$. Please suggest how I should proceed. Thanks!
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0answers
28 views

Explicit matrix representation of an algebraic extension

This may be considered an extension of this question. Let $\mathbb{F}$ be a field, and let $p(X)\in\mathbb{F}[X]$ be an irreducible polynomial. Let $\mathbb{F}_p$ be the extension of $\mathbb{F}$ by ...
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0answers
20 views

Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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1answer
39 views

Elements in a Field of size $27$

I constructed the Field $$F_3[x]/<1 + 2x + x^3>$$ as the question asked to construct a field of size $27$ and I understood everything up to this point. The solution then says the elements in ...
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0answers
11 views

On the question of the Galois group of some polynomial. [duplicate]

I want to ask you some question on the Galois group of some polynomial. Let $p_1<p_2<\cdots<p_n$ be distinct prime numbers. Let's consider $f(t)=(t^2-p_1)(t^2-p_2)\cdots(t^2-p_n)\in \...
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1answer
42 views

Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
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0answers
54 views

What does the irreducible polynomial over $\mathbb{Q}$ and $\mathbb{Q}(i)$ mean?

So here'as the problem Find the monic irreducible polynomial $g(x) \in \mathbb{Q}(x)$ for $i+ \sqrt{3}$ over $\mathbb{Q}(i)$ Erm...huh? Okay, so a minimal polynomial $m$ is what I need, and ...
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1answer
33 views

Algebraic function fields

I am trying to understand what an algebraic function field is, so i was looking for some examples. The example on Wiki says: Given a polynomial ring $k[X,Y]$. Consider the ideal generated by the ...
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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)...
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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 ...
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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 ...
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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 ...
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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 ...
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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[...
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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 $\mathbb{F}_q[x, \sqrt{f}]$....
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votes
1answer
52 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 ...
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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^...
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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}(...
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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 ...
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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 ...
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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(...
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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 ...
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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 $L$ ...
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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 ...
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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
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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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2answers
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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
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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}(...
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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
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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 $...
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2answers
49 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}(\...
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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 = ...
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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 ...
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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 ...
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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 $\mathcal{O}_F$?
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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 ...
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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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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
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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, \...