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 ...

learn more… | top users | synonyms (1)

1
vote
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
63 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)? ...
-1
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
51 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 ...
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
107 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 ...
0
votes
1answer
22 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
18 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
25 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
73 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
47 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 ...
1
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
36 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
50 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 ...
0
votes
0answers
16 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
57 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 ...
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 ...
2
votes
1answer
35 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
48 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 ...
1
vote
0answers
37 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
64 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 $S\cong A/I$ ...
2
votes
2answers
29 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
0
votes
1answer
61 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 ...
1
vote
1answer
55 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 ...
2
votes
3answers
70 views

$\mathbb{R}[x]/\langle x^2+1\rangle$ and $\mathbb{R}[x]/\langle x^3 +x+1\rangle$ are isomorphic?

$\mathbb{R}[x]/\langle x^2+1\rangle$ and $\mathbb{R}[x]/\langle x^2 +x+1\rangle$ are isomorphic or not? I guess these are isomorphic as they are isomorphic to the field of complex number. But how can ...
3
votes
1answer
51 views

Why do we need the infinite field hypothesis in this cohomology calculation?

I've just finished my very first calculation with sheaf cohomology. It's exercise III.2.1(a) in Hartshorne, and it says Let $X = \mathbb{A}_K^1$ be the affine line over an infinite field $K$. Let ...
3
votes
2answers
51 views

$[k(\alpha):k]=p, [k(\beta):k]=q$, $p>q$ are primes, then $k(\alpha,\beta)=k(\alpha+\beta)$

Let $p>q$ be primes. Suppose $L\mid_{k}$ is an algebraic extension and $\alpha,\beta\in L$ are such that $[k(\alpha):k]=p$, $[k(\beta):k]=q$ and characteristic of $k$ is coprime with $p$. Show that ...
-1
votes
1answer
27 views

Finite fields and generators of Galois group [closed]

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
1
vote
1answer
42 views

If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{{a_1}^p,\ldots,{a_n}^p \}$

Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent ...
1
vote
1answer
29 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If ...
2
votes
1answer
9 views

If $K/F$ is algebraic and $a\in K$ is separable over $F(a^p)$ then $a\in F(a^p)$

Suppose that $F$ is a field of characteristic $p$. Show that if $K/F$ is algebraic and $a\in K$ is separable over $F(a^p)$ then $a\in F(a^p)$. I know that the minimal polynomial of $a$ has ...
4
votes
0answers
55 views

When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
0
votes
1answer
42 views

The fixed field of $A$ is equal to the fixed field of $\langle A\rangle$.

Let $E$ be a finite extension field of $F$. Let $A$ be a subset of $Gal(E/F)$. Let $\langle A\rangle$ be the subgroup generated by $A$. Is the fixed field of $A$ equal to the fixed field of $\langle ...
0
votes
0answers
33 views

Is $D$ a field?

Problem. Let $D$ be an integral domain and let $\mathcal{F}(D)$ be a field of quotients of $D$. If $D\subset \mathcal{F}(D)$ then prove or disprove that, $D$ is a field. ...
0
votes
0answers
18 views

If K is a field whose characteristic is not 2, show F/K is Galois [duplicate]

Let F/K be a field of extension 2, If K is a field whose characteristic is not 2, show F/K is Galois. I think I need to use a fact that the extension F/K is Galois if and only if K is the splitting ...
0
votes
0answers
35 views

Splitting field of a set of polynomials

Given $X\subseteq F[x]$ where $F$ is a field, how to prove that there exist a splitting field of $X$ over $F$? In the case that $X$ is finite, I think the answer can be solved using Kronecker's ...
2
votes
1answer
40 views

Is there a systematic way of finding the factorization over the closure of $\mathbb{Z}_2$ of $p(x) = x^{32} - x$?

And similar polynomials of the form $x^{p^n} - x$. I know that the degrees of the irreducible monic polynomials that factorize $x^{32} - x$ will have degree $d \vert 5 = 1, 5$. Also, I know that $x$ ...
0
votes
1answer
38 views

Find basis in Extension field

I want to know that if we are asked to find the minimal polynomial, what are the steps? So if $F$ is a field and $\alpha$ is algebraic over F, first we need to find $[F(\alpha):F]$ and then ...
-1
votes
0answers
14 views

CharK=0 (or p) iff CharF=0 (or p), F is subfield of K [duplicate]

Let $F$ be a subfield of the field $K$. Prove that: 1) $CharK=0 \iff CharF=0$ 2) $CharK=p \iff CharF=p,\ p$ is prime. My thoughts: (a) $1_K \in K$, so $ CharK=ord(1_K) \ | \ |K|$ from Lagrange. If ...
1
vote
1answer
21 views

Order of an orbit of Frobenius action on a algebraically closed field of characteristic p

Consider the action of the Frobenius homomorphism $F^{2}:\,\overline{\mathbb{F}_{q}}\rightarrow\overline{\mathbb{F}_{q}},\,x\rightarrow x^{q^{2}}$ over $\overline{\mathbb{F}_{q}}$ . Let $s=\left\{ ...
0
votes
0answers
25 views

Complex extension isomorphic to $\mathbb{R}$?

Let $K$ be some field extension of $\mathbb{Q}$ containing some complex number $c=a+bi$ with $a,b\in \mathbb{R}$ and $b\neq 0$. Is it possible that $K\cong \mathbb{R}$ as fields? I tried to disprove ...