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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0
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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 ...
1
vote
0answers
21 views

Problem when computing ideal class group

When computing the ideal class group of a quadratic extension $\mathbb{Q}[\alpha]$ after we have decomposed all rational primes smaller than the Minkowski bound into generating prime ideals ...
4
votes
0answers
30 views

The set of zeros of some polynomial over the closure of a finite field constitute a field in its own right

Let $p$ be a prime and $n$ be a positive integer and let $p(x) = x^{p^n} - x$ be a polynomial in $\mathbb{Z}_p[x]$. Let $Q$ be the set of all zeros of $p(x)$ over the algebraic closure of ...
-1
votes
1answer
46 views

Find the minimum polynomial of $u$ over $Q$ where $u=\sqrt3-(1+(5/2)^{1/3})^{1/4}$ [closed]

I tried using the binomial theorem but the terms keep increasing indefinitely
0
votes
1answer
16 views

Center of finite dimensional division $\mathbb{R}$-algebra?

Let $D$ be a finite dimensional division $\mathbb{R}$-algebra. Why is it that $Z(D)=\mathbb{R}$ or $Z(D)=\mathbb{C}$? I have seen an explanation: It is because $\mathbb{C}$ is the only non ...
0
votes
0answers
22 views

What does "class $x \in X$ mean?

I'm working through Escofier's book on Galois Theory, and in several exercises regarding finite fields they use the terminology "class $x \in X$, and I'm not certain what it means. For instance, let ...
0
votes
1answer
16 views

Problem on field extension related to irreducible polynomial

Suppose $\gamma,\gamma'\in\Bbb C$ are distinct roots of the same irreducible polynomial $p\in\Bbb Q[x]$. Suppose $x^2-5$ is irreducible in $\Bbb Q(\gamma)[x] $. Show that it is also irreducible in ...
1
vote
1answer
43 views

Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
0
votes
1answer
54 views

13th root of 2 in field $\mathbb{F}_{13}$ [closed]

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
0
votes
1answer
41 views

Show that $\operatorname{Gal}(K/\mathbb Q)$ can be identified with the set of embeddings of $K$ into $\mathbb C$

I would be grateful if someone could help me demonstrate the following easy fact. Let $K$ be a number field which is Galois over $\mathbb Q$ and $\tau_0:K\hookrightarrow \mathbb C$ a fixed $\mathbb ...
0
votes
2answers
61 views

Isomorphic subfields of $\mathbb C$ [closed]

Sorry if this is a very trivial question but I can't find a proof or a counterexample to it. If $K$ and $L$ are isomorphic subfields of $\mathbb C$ both containing $\mathbb Q$ then are they identical ...
0
votes
2answers
40 views

Computing the degree of the splitting field of $x^3+18x+3$ over $\Bbb Q$

Let $T$ be a splitting field of polynomial $$f(x)=x^3+18x+3\in\mathbb{Q}[x].$$ What is the degree $[T:\mathbb{Q}]$? My thoughts: the polynomial $f$ is irreducible over $\mathbb{Q}$, therefore the ...
1
vote
1answer
50 views

Isomorphic Galois groups imply isomorphic field extensions?

Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also ...
1
vote
0answers
28 views

All possible degree 3 field extensions

Do we know anything about all possible degree 3 field extensions? If characteristic is 3: If Galois, then Artin-Schreier. If inseparable, then this is just cube root. If separable, but not normal, ...
2
votes
2answers
61 views

How to read this problem from Dummit-Foote's “Abstract Algebra”?

Problem 14.6.1 on page 617 says Show that a cubic with a multiple root has a linear factor. Is the same true for quartics? Let $f \in F[x]$ be a cubic. If $f$ has a root in $F$, let alone a ...
0
votes
0answers
24 views

Minimal polynomials over an inseparable extension

Let $F$ and $K$ be fields, and $\sigma$ and $\tau$ be two maps between them, $F \underset{\sigma}{\overset{\tau}{\rightrightarrows}} K$. Let $\alpha$ be an element algebraic over $F$, with minimal ...
1
vote
0answers
43 views

“Closure” of a polynomial ring by fraction field

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular noetherian $k$-algebra, $K$ the fraction field of $A$ and $\bar{K}$ an algebraic closure of $K$. Does there exist a ...
1
vote
2answers
26 views

Is there a way to see if $\alpha \in \mathbb{C}$ is constructible at a glance?

The notion of constructibility is not too obscure but mathematically, I find the definitions tedious and not very easy to handle with. I don't know if Ian Stewart's book Galois Theory edition 4 ...
2
votes
2answers
31 views

What is the definition of “prime ideal decomposition”?

I'm reading about Sunada's theorem in the book Geometry and Spectra of Compact Riemann Surfaces (Peter Buser) and I encountered this paragraph: If R is an algebraic number field and if $p \in ...
2
votes
3answers
44 views

$\mathbb{Z} [\sqrt{2}]$ is an integral domain

We know that $(\mathbb{Z} [\sqrt{2}],+,\cdot)$ is an integral domain. Someone can prove it easily if he says that is a subring of $(\mathbb{R} ,+,\cdot)$ . Can we find a different proof, more ...
0
votes
1answer
33 views

The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
3
votes
0answers
44 views

Kähler differentials in an inseparable field extension

Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. ...
0
votes
0answers
46 views

Prove that $\sin ^{-1} 1 $ is algebraic over $\mathbb Q$ [duplicate]

Prove or disprove the following : $1.\sin ^{-1} 1 $ is algebraic over $\mathbb Q$ $2.\cos (\frac{\pi}{17})$ is algebraic over $\mathbb Q$ As suggested by @Andre ,for the 2nd one ...
0
votes
0answers
12 views

Algorithm for ordering on an algebraic number field

Given an algebraic field extension of the rationals $Q(P(X))$, where $P(X)$ is a polynomial in $X$, how do I algorithmically define an ordering on $Q(P(X))$ that is compatible with a specific real ...
4
votes
0answers
56 views

Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
1
vote
1answer
46 views

If $x+y\sqrt{n} \in \mathbb{C}$ is a root of $f$ then $x-y\sqrt{n}$ is also a root

Let $n\in \mathbb{Z}$ be a non-square integer and $x+y\sqrt{n} \in \mathbb{C}$ a root of $f\in \mathbb{Q}[x]$ with $x,y\in \mathbb{Q}$. Show that $x-y\sqrt{n}$ is also a root of $f$. To show ...
0
votes
1answer
26 views

Algebraically closed field and polynomials

There is the problem: Let $F$ be a field of characteristic $0$ with the condition: If $f(x) \in F[x]$ has no roots in $F$, then the degree of $f(x)$ is a multiple of $21$. Prove that $F$ is ...
1
vote
2answers
33 views

How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
2
votes
1answer
49 views

Additive combinatorics and $|S \cap (S+t)| \geq K |S|$ over $\mathbb{F}_2^n$

I don't know much about additive combinatorics, and I am wondering if there are results concerning the size of $|S \cap (S+t)|$ where $S \subset \mathbb{F}_2^n$ and $t \in \mathbb{F}_2^n$. Especially, ...
1
vote
0answers
9 views

Finding the roots with the largest magnitude

Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with ...
5
votes
0answers
79 views

Show that a sum of consecutive radicals is irrational $\forall n$ [closed]

I need to show that the number $\sqrt 2+ \sqrt[3]{3}+\sqrt[4]{4}+\sqrt[5]{5}+\cdots+\sqrt[n]{n}$ is irrational for any $n\ge2$. I don't have a clue about how I could show that. Thank you!
-3
votes
1answer
37 views

Find the degree of the splitting field of a polynomial over $Q$ [closed]

I want to know how to determine the degree of the extension $K/\mathbb{Q}$, where $K$ is the splitting field of the polynomial $x^6+1\in \mathbb{Q}[x]$ over $\mathbb{Q}$. Do I have to get all the ...
1
vote
0answers
40 views

Find the value of $n$

Let $F$ be a field having $5^n$ elements .Also $F$ has an element which satisfies $x^{5^n}=1$ such that $x\neq 1$. Find $n$ . My try: Let $x\in F $ satisfy $x^{5^n}=1$ .Obviously the group ...
0
votes
1answer
35 views

Find the lattice of Galois Field

I am wondering what the lattice of subfield of $GF(p^{30})$ looks like. I know that it starts from $GF(p)$ and then $GF(p^2)$ and $GF(p^3)$, but then I am lost. And I looked it up online, but can't ...
2
votes
1answer
19 views

non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
1
vote
1answer
70 views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] ...
0
votes
1answer
27 views

Show that any field K has a subfield isomorphic to either $\mathbb{Q}$ of $\mathbb{Z}_p$

Show that any field K has a subfield isomorphic to either $\mathbb{Q}$ of $\mathbb{Z}_p$ I understand that here we are talking about a prime subfield that would be isomorphic to either one or the ...
1
vote
2answers
28 views

Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
2
votes
2answers
38 views

Galois group of a quartic which is also a quadratic in $x^2$

A few weeks ago a professor of mine mentioned that the Galois group of a certain type of quartic polynomial is easy to calculate, and at the time it seemed obvious to me so I didn't ask why. Now i'm ...
2
votes
0answers
30 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
2answers
31 views

Degree of splitting field of $x^6-2$ over $\mathbb{Q}$

Let $f=x^6-2$, find the degree of splitting field of $f$ over $\mathbb{Q}$. I calculated the roots of $f$ are $\pm \sqrt[6]{2},\pm e^{i \pi/3}\sqrt[6]{2},\pm e^{2i \pi/3}\sqrt[6]{2}$. I suspect ...
1
vote
1answer
37 views

how would you show that field automorphisms fix prime subfields?

Suppose K is a prime subfield of E, then if $\phi$ is an automorphism from E to E, we have for all x $\in$ K, $phi(x) = x$. I feel like this is just the definition of a field automorphism, but my ...