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 ...

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1answer
37 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
2
votes
1answer
65 views

Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
0
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0answers
13 views

Transcendental field extensions obtained by taking quotient of $k[X_1, \ldots, X_n]$

Given a field $k$, $n \in \mathbb{N}\setminus \lbrace 0 \rbrace$ and $M$ a maximal ideal of $k[X_1, \ldots, X_n]$, can the field $L = k[X_1, \ldots, X_n]/M$ ever be transcendental over $k$? By this I ...
2
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1answer
23 views

Which cyclotomic fields are different?

For $n$ a positive integer, let us write $\zeta_n = e^\frac{2 \pi i}{n}$, a primitive $n$th root of unity. It is clear that, if $m$ divides $n$, then we have an inclusion of cyclotomic fields $$ ...
0
votes
1answer
14 views

Systematic way of expressing field extensions

If a field $Q$ were to be extended to include roots of the quadratic polynomial $x^2$$-2=0$, the extended field $Q$($\sqrt2$) would include elements of the form $a$ + $b$$\sqrt 2$. However, extending ...
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votes
1answer
28 views

Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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0answers
35 views

does isomprphic fields have exactly the same properties?

It is written in many books that isomorphic fields have exactly the same properties. Does that mean only to the algebraic properties (i.e. properties that derived from the field operations)? To ...
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0answers
9 views

When does a system of n symmetric polynomials in n variables have exactly one solution over C up to permutation?

I was slightly amused that if I never learned about polynomials and was asked if Vieta's system of equations has exactly one solution up to permutation, the solution would be to develop polynomials in ...
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3answers
27 views

What are the ideals of $F_2[x]/\langle x^2 + x +1\rangle$? [closed]

Is it just the divisors of $x^2 +x+1$ in mod $2$ ?
2
votes
2answers
64 views

The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
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3answers
24 views

Ring Extension: Mapping: $ \mathbb Q[\sqrt d] \rightarrow \mathbb Q$

Show that the Norm: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q, (r+s\sqrt d) (r-s\sqrt d) = r^2 - ds^2$ is multiplicative, i.d. $N(xy) = N(x)N(y)$ How to show it without computing? (I tried to do it ...
0
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1answer
30 views

A problem about degrees of minimal polynomials for two arbitrary elements in an extension field

I'm struggling to come up with a reasonable proof for the following problem: Suppose $E$ is an extension field of a field $K$ and that $a$ and $b$ are algebraic elements in $E$. Show that the ...
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2answers
37 views

Extension fields, and their cardinality and roots

I have no idea how to begin answering this question. My notes do not help. Let $f(X) = X^3$ + $X + 1 ∈ \mathbb Z_5[X]$ ($\mathbb Z_5$ denotes the integers mod 5). Let $E=\mathbb Z_5[X]/(f(X))$. ...
1
vote
1answer
25 views

Show that $f$ is the minimal polynomial of $u$

Let $u$ be a root of $f=x^3-x^2+x+2\in \mathbb{Q}[x]$ and $K=\mathbb{Q}(u)$. Prove that $f=m_\mathbb{Q}(u)$. I have no idea how to approach this problem. Should I prove that $f$ is irreducible ...
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votes
2answers
38 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
0
votes
1answer
15 views

Unclear explanation of solution again;field extension

The solution sheet assumes additional knowledge than what is provided, which annoys me; I don't understand this. Here's the problem $L:K$ is a field extension. If $\alpha,\beta \in L$ is ...
1
vote
1answer
18 views

why aren't finite fields of prime characteristic algebraically closed?

How can this be proven? I know that if a field has a prime characteristic, any element of the field, say $a$. will satisfy the following equation: $ap = 0$, where p is the prime characteristic of ...
1
vote
0answers
30 views

Since field extensions are vector spaces over ground fields, how is linear algebra used to study them?

I've already taken a semester of Galois theory, so I'm not asking about anything that would be covered there, like the tower theorem. Specifically, multiplication by an element of a field extension is ...
3
votes
2answers
46 views

Galois group of splitting field over $\mathbb{Q}$

Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$ I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the ...
2
votes
1answer
24 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
2
votes
0answers
40 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
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0answers
24 views

The tower of ramification indices

Let $K\subset L\subset M$ be an extension of number fields. Let $R\subset S\subset T$, be their algebraic integers rings, respectively. Suppose that $P\subset R$, $Q\subset S$, $U\subset T$ be a prime ...
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votes
1answer
30 views

Prove a field - trouble with defining basic operations.

I'm certain this is a fairly easy question, but my algebra is rusty and I'm doing this as a part of a bigger proof. I'm stating that, if $\Bbb K$ is a field and $\Bbb K'$ its prime subfield, then 1) ...
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0answers
18 views

When are composite extensions isomorphic?

Let $E$ and $F$ be two totally complex finite extensions of $\mathbb{Q}$, let $\sigma_i \, :\,E \rightarrow \mathbb{C}, i\in I$ and $\tau_j \,: \,F \rightarrow \mathbb{C}, j\in J$ denote all their ...
2
votes
1answer
111 views

Splitting field of an irreducible polynomial of degree four [on hold]

Suppose $f$ is the irreducible polynomial $x^4+x+1$ in $\mathbb{Q}[X]$ and we will call $E_f$ the splitting field of $f$ (which is a subfield of $\mathbb{C}$). Suppose $\alpha$ is a complex root of ...
0
votes
2answers
21 views

Abstractly constructing splitting fields

I have a series of exercises where I have to determine the degree of various splitting fields. I am freely using the following observation, which I feel is intuitively true, but I am asking here to ...
1
vote
1answer
26 views

An algebraic element $a$ in a field extension $K/F$ satisfies $a^{q^m}=a$

Let $F$ be a field with order $q$ and characteristic $p$. Show that if $a$ is an algebraic element over $F$ in the extension $K$, then $a^{q^m}=a$ for some $m$. I have shown that the order of the ...
0
votes
0answers
23 views

How can one show algebraically that an angle is constructible?

For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. ...
1
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0answers
20 views

$f(x) = x^2 + bx + a$ irreducible over $\Bbb F_p$ (finite field of $p$ prime elements) iff $(b^2 - 4a)^{\frac{p-1}{2}} = -1$ in $\Bbb F_p$

My attempt started as follows. I know that for $f$ to be irreducible, $D = b^2 - 4a$ is not a square in $\Bbb F_p$ (ie $(\frac{D}{p}) = -1$). I also know that $D^{p-1} = 1$, so I see $\sqrt{(D^{p-1})} ...
1
vote
0answers
32 views

Structure of Galois group

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$ and $K|_F$ be the splitting field of $f(x)$. I want to prove the statement that if $G = \text{Gal}(K|_F)$ is cyclic then $[K:F] ...
1
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1answer
33 views

$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
0
votes
1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
2
votes
2answers
46 views

Proving $f(x)$ is not a square in $k[x]$

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
1
vote
1answer
39 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
4
votes
5answers
85 views

Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$

Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ and find all $w\in \mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ such that $\mathbb{Q}(w)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. It ...
4
votes
1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
0
votes
1answer
23 views

Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
2
votes
2answers
58 views

$\operatorname{char}R=0 \implies\mathbb{Q} \hookrightarrow R$

Let $R$ be any field, then: $$\operatorname{char}R=0 \implies \mathbb{Q} \hookrightarrow R$$ Proof: We know that $\mathbb{Q} = Q(\mathbb{Z})=\{[(x,y)]\subseteq\mathbb{Z}\times \mathbb{Z^*}:(x,y) ...
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2answers
46 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
0
votes
2answers
35 views

Splitting field of $x^5-3x^3+x^2-3$

I am trying to solve the following problem, Find the degree of the splitting field of the polynomial $p(x)=x^5-3x^3+x^2-3$ over $\mathbb{Q}.$ My approach for solution: Clearly -1 is a root of the ...
2
votes
2answers
42 views

If Q(a,b) is a field extension, can we always choose an equivalent extension Q(c) such that c=a+b?

If we have two complex numbers $a,b$ that are algebraic over $\mathbb {Q} $, we can make an extension $\mathbb {Q}(a,b)$ that is equal to an extension $\mathbb {Q}(c)$ for some $c\in \mathbb {C} $. ...
3
votes
0answers
31 views

Infinite extensions of “finite degree under $\mathbb{Q}$” [duplicate]

Consider an algebraic extension $K$ of $\mathbb{Q}$. The degree $[K:\mathbb{Q}]$ of $K$ is defined as the dimension of the extension considered as a vector space. Now, let $\overline{\mathbb{Q}}$ be ...
0
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0answers
23 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
0
votes
0answers
16 views

Frobenius Map and Subfields of $\bar{\mathbb{F}}(x,y)$

Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) ...
1
vote
1answer
37 views

Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
1
vote
1answer
41 views

$\mathbb{Z}_p \hookrightarrow R$ is it necessary $R$ commutative?

Let $R$ be an integral domain with $\operatorname{Char}(R)=p$, with $p$ prime. Then: $$\mathbb{Z}_p \hookrightarrow R$$ The proof is not difficult. My questions are: 1) Is it necessary to have an ...
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votes
1answer
43 views

$X^4-10X+1$ reducible in $\mathbb{F}_p[X]$ for all prime $p$ [duplicate]

Show that the polynomial $X^4-10X+1$ is irreducible in $\mathbb{Z}[X]$ but reducible in $\mathbb{F}_p[X]$ for all prime $p$. I could show the irreducibility in $\mathbb{Z}[X]$ but not sure how to ...
2
votes
0answers
28 views

char$(K)=0$ then $K(x^{2}) \cap K(x^{2}-x)=K$ [duplicate]

Let $K$ be a field of characteristic zero and $K(x)$ the field of rational functions with coefficients in $K$. Let $K(u)$ denote the subfield of $K(x)$ generated by $u \in K(x)$ over $K$. My ...
3
votes
2answers
58 views

Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
6
votes
1answer
148 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...