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....

learn more… | top users | synonyms (1)

3
votes
0answers
112 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
3
votes
0answers
63 views

$\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism $\...
3
votes
0answers
97 views

Proving Lagrange's Theorem with Galois Theory

Problem: Let $G$ be a finite group with subgroup $H$. Let $|G| = n$ and $|H| = m$. Prove that the order of $H$ divides the order of $G$ using only results from Field and Galois Theory, with the ...
3
votes
0answers
105 views

Factoring irreducible polynomial over normal extension

Let $f$ be an irreducible polynomial over $F$ and $K/F$ be a normal extension. How to prove $f$ is factored by product of irreducible poly. over $K$ with same degree? I tried to do it by if $f_1, ...
3
votes
0answers
250 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly infinite)...
3
votes
0answers
482 views

Normal closure of field extension, axiom of choice

Update My previous proof was incorrect. This updated proof is inspired by the comment by 'MartianInvader'. Problem I can prove the statement 'Every algebraic extension $L:K$ has a normal closure $F:L$...
3
votes
0answers
97 views

Is there a more efficient way of proving a polynomial is primitive?

I have found an impressive Java implementation of a $ℤ2$ Polynomial class in which there seems to be an efficient implementation of a test for reducibility (see <...
3
votes
0answers
66 views

Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
3
votes
0answers
259 views

Non-isomorphic simple extensions of the same degree of a field of positive characteristic

Let $K$ be a field of positive characterstic. I need to show that there exists $a$ and $b$ of the same degree over $K$ but $K(a)$ and $K(b)$ are not isomorphic. I thought of an example where they are ...
3
votes
0answers
187 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
3
votes
0answers
140 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
3
votes
0answers
161 views

Fixed points of automorphism in the field $\mathbb{C}(x,y)$

I am trying to solve a problem, and one of the parts is the following: Let $M=\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ be a non singular $2\times 2$ matrix with integer ...
3
votes
0answers
119 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of $\alpha$...
3
votes
0answers
97 views

The Galois group of $f(t)=t^3-x_1t^2+x_2t-x_3$

I'm trying to solve this question: Let $F[x_1, x_2,x_3]$ be a polynomial ring in $x_1, x_2,x_3$ over a field $F$. Let $K=F(x_1,x_2,x_3)$ be the field of rational functions (i.e., the ...
3
votes
0answers
144 views

When the extensions Q(α,β) and Q(αβ) over Q are the same?

I would like to know in which conditions the extensions $\mathbb{Q}(\alpha,\beta)$ and $\mathbb{Q}(\alpha\beta)$ over $\mathbb{Q}$ are the same. Thanks in advance.
3
votes
0answers
950 views

Quick explanation: calculating class group of $\sqrt{-21}$

I'm trying to calculate the class group of $\mathbb{Q} (\sqrt{-21})$, and I've managed to confuse myself. I've used the standard Minkowski bound to show that we only need to consider principal ideals (...
3
votes
0answers
399 views

field embedding

I've come across this problem in Etingof's notes on representation theory (Problem 5.1 on p. 78). It just sounds nice exercise... The question is : Let $f : k(x_1,\ldots,x_n)\rightarrow k(y_1,\...
2
votes
0answers
51 views

Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
2
votes
0answers
37 views

Isomorphism of transcendental extensions

If $a,b$ are transcendental over $\mathbb{Q}$, then it is known that $\mathbb{Q}(a)$ and $\mathbb{Q}(b)$ are isomorphic. Consider a simple case: suppose $a,b,c$ are transcedental over $\mathbb{Q}$. ...
2
votes
0answers
23 views

Proof verification: Show that the Frobenius map is surjective.

I would like to prove the following but I would like someone to check my proof. For an algebraically closed field $K$ with characteristic $p$, the Frobenius map $F(x) = x^p$ is surjective What I ...
2
votes
0answers
29 views

Galois Group Solution Check

Find the Galois group $G = \text{Gal}(\mathbb{Q}(\omega_{12})/\mathbb{Q})$, and its lattice of subgroups, where $\omega_{12}$ is the 12th root of unity. We have that $$\text{Gal}(\mathbb{Q}(\omega_{...
2
votes
0answers
42 views

Field extensions that decompose into towers of degree$\leq n$ extensions

Let $F$ be a field and let $n$ be a natural number. Consider the class of field extensions $E/F$ that decompose into towers $E=E_k/E_{k-1}/\cdots/E_1/E_0=F$ such that $[E_{i+1}:E_i]\leq n$ for $i=0,1,...
2
votes
0answers
53 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 $...
2
votes
0answers
33 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
2
votes
0answers
77 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 ...
2
votes
0answers
45 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 $F$) ...
2
votes
0answers
19 views

Automorphisms of field generated by two coprime elements

I would like to know if the follwing statement is true: Let $F$ be a field and let $a,b$ be algebraic over $F$ with coprime degrees $m$ and $n$, respectively. Suppose $F(b)/F$ is normal. Letting $K = ...
2
votes
0answers
22 views

On the restriction of field homomorphisms to subfields

Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace \alpha_1,\dots,\...
2
votes
0answers
21 views

Quick help on why this extension is of degree $2$

The set up is, Splitting field $K=\mathbb{Q}(i,\alpha)$ where $\alpha=2^{\frac{1}{4}} \in \mathbb{R}$. The three obvious subfields of $K$ with degree $2$ over $\mathbb{Q}$ are..? Answer is ...
2
votes
0answers
22 views

Maximum ideal in field

Let $k$ be a field, $n ∈ \mathbb{Z}>0$, and $α_1, α_2, ..., α_n ∈ k$. Prove: $(x_1 − α_1, x_2 − α_2, \ldots, x_n − α_n)$ is a maximal ideal. I cannot figure out how to prove this; what is meant ...
2
votes
0answers
38 views

Proof that $\mathrm{Aut}(K\left[ \sqrt[m]{g \in K} \right])/\mathrm{Aut}(K) = \mathbb{Z}_w, w | g $

Let $K$ be an extension field of $\mathbb{Q}$. That is $$ K = \mathbb{Q}[r_1,\ldots,r_k ]$$ I am considering $\mathrm{Aut}(K)$ which is the group of field automorphisms of $K$ and I wish to show ...
2
votes
0answers
26 views

Transcendental extension not isomorphic to its closure

Suppose I'm given a field extension $K/F$ with $\alpha\in K$ transcendental over $F$, the claim is that $F(x)\cong F(\alpha)$. It's a statement without proof in our class notes, and the remarks ...
2
votes
0answers
67 views

What is the number of subgroups of $C_2 \times C_2 \times C_2 \times \cdots \times C_2$?

I think that counting the number of subgroups of various groups is usually very difficult. I was wondering about the number of subgroups of $(C_2)^n$. For example, there are 5 subgroups of $C_2 \...
2
votes
0answers
49 views

Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the "...
2
votes
0answers
50 views

I don't understand what a Pythagorean closure of $\mathbb{Q}$ is; how are these definitions equivalent?

I have two definitions of the said field. And frankly I don't see why one is equivalent to the other. It just doesn't add up. Let's look at wikipedia's definition. In algebra, a Pythagorean field ...
2
votes
0answers
48 views

Proving that primes of the form $4k + 1$ are not Gaussian primes

Let $p$ be a positive integer prime such that $p\equiv1\pmod4$. Assume that $p$ is a Gaussian prime, so $(p)$ is a maximal ideal of $Z[i]$, and consider the field $K = Z[i]/(p)$. We state without ...
2
votes
0answers
47 views

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
2
votes
0answers
48 views

Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
2
votes
0answers
29 views

How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
2
votes
0answers
27 views

Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $ < \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
2
votes
0answers
93 views

How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt ...
2
votes
0answers
40 views

Show that $|\text{Hom}_k(K,\widetilde{K})(\phi)|\le [K:L]$

Let $K/k$ be a finite field extension, $L$ an intermediate field and $\widetilde{K}$ such that $\widetilde{K}/k$ is normal. Let $\phi \in \text{Hom}_k(L,\widetilde{K}) := \{\psi:L\rightarrow \...
2
votes
0answers
40 views

Galois extension of $\mathbb C$ is also Galois over $\mathbb R$??

Is any Galois extension of $\mathbb C$ also Galois over $\mathbb R$? I know if that extension is finite, Then it is true because $\mathbb C$ is algebraically closed. But How about the infinite case? ...
2
votes
0answers
147 views

Is the splitting field of $g=x^3-3x-1$ over $\Bbb Q$ a radical extension?

Since $g=x^3-3x-1$ is irreducible over $\Bbb Q$, and has square discriminant. If $L$ is the splitting field of $g$ over $\Bbb Q$, since $g$ has square discriminant we have $\text{Gal}(L/\Bbb Q) = ...
2
votes
0answers
35 views

Showing that $|\text{Hom}_\mathbb{Q}(K,K)|=6$

Let $g\in \mathbb{Q}[X]$ be irreducible with $\text{deg}\;g=3$. We assume that only one of the roots of $g$ is in $\mathbb{R}$. So, $\alpha \in \mathbb{R}$. Let $L\subset\mathbb{C}$ be the splitting ...
2
votes
0answers
38 views

Prove that $F$ is finite field

We denote $\mathbb{F}_q$ as finite field of $q$ elements and the algebraic clausure as $\overline{\mathbb{F}}_q$. Let $$K=\bigcup_{n=1}^{\infty}\mathbb{F}_{q_n}\subset \overline{\mathbb{F}}_3,\; \...
2
votes
0answers
44 views

To prove a equality of field norm by field extension.

We know that if we set $K$, $F$ and $L$ fields, with $L$ a finite extension of $F$ and $F$ a finite extension of $K$. Then we have the norm equality $N_{L/K}=N_{F/K}N_{L/F}$. The common solution is by ...
2
votes
0answers
50 views

Connection between Algebraic structures and Formal Systems

I am a college student with very little mathematics background (up to Calculus 152 at Rutgers University), but have become increasingly interested in computing and mathematics in the last year. I am ...
2
votes
0answers
13 views

Show that $K^\sigma $ is a field, and some other properties.

Let $K$ a field and $\sigma \in Aut(K)$ (the set of automorphism $K\to K$). Let denote $K^\sigma =\{x\in K\mid \sigma (x)=x\}$ the set of fix point. 1) Show that $K^\sigma $ is a subfield of $K$. 2)...
2
votes
0answers
68 views

Give an example of a finite extension of fields that is neither separable nor normal.

The title says it all. This is not homework. Haven't made much progress - I know that with $K=F_2(t)$, $L=K(\sqrt t)$ (i.e. adjoining the square root of $t$ to the function field with coefficients in ...