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
45 views

Give an example of a polynomial whose Galois group is isomorphic to $D_{10}$

I have been spending my leisure time determining the subfield lattices and corresponding Galois subgroup lattices of some splitting fields of polynomials. I have made the lattice diagrams for the ...
2
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0answers
71 views

Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
11
votes
1answer
119 views

Surjective exponentials for algebraically closed fields

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ ...
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2answers
27 views

For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
2
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3answers
74 views

Showing $\mathbb{Q} \times \mathbb{Q}$ is not a field

I am revising and have come across the question Show that $\mathbb{Q} \times \mathbb{Q}$ with element-wise addition and multiplication is not a field I don't understand how to go about this, do i ...
2
votes
2answers
69 views

Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
3
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1answer
35 views

On the degree of cyclotomic fields extension.

Let $F$ be a field and $n$ be any integer satisfying $(n,\text{char}(F))=1$ when $\text{char}(F)\ne 0$. Let $\xi$ be a primitive $n$-th root of unity. I know that $\text{Gal}(F(\xi)/F)$ is isomorphic ...
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1answer
35 views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
2
votes
1answer
44 views

Determine the degree of an extension field over $\mathbb{Q}$

Let $\alpha = e^{\frac{i\pi}{6}}$. Compute $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and find the minimal polynomial of $\alpha$, $m_{\mathbb{Q}}(\alpha)$. I can see clearly that $\alpha^6+1=0$ but I ...
4
votes
1answer
41 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
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2answers
41 views

Confusion about elements in fields, like -1 in Z5

I'm learning field and ring theory, and I've repeatedly seen the usage of -1, -2 and -3 as elements of $\mathbb{Z}_5$. As far as my knowledge goes, $\mathbb{Z}_5$ consists of {0,1,2,3,4}. This is ...
4
votes
1answer
62 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
0
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1answer
39 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
70 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
15 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
24 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
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1answer
15 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 ...
0
votes
1answer
30 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
36 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 ...
0
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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
73 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
25 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
31 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
39 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
30 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 ...
0
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
16 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
20 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
33 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
49 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
26 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
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0answers
44 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
31 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 ...
0
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
19 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 ...
1
vote
1answer
120 views

Splitting field of an irreducible polynomial of degree four [closed]

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
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1answer
28 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 ...
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0answers
25 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. ...
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0answers
23 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
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0answers
34 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
34 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
47 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
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1answer
45 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
87 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 ...
0
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0answers
25 views

If $L=K(a_1,…,a_n)$ can we find an irreducible polynomial in $K[x]$ s.t. $p(a_i)=0$ for all $i$?

I have the following question that I can't prove or find a counterexample for. Let $K$ be a field and $L$ a finite field extension of $K$ so that we can write $L=K(a_1,..,a_n)$ where all $a_i\in ...
4
votes
1answer
139 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
24 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 ...