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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6
votes
1answer
128 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
1
vote
0answers
14 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, ...
1
vote
1answer
12 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 ...
2
votes
2answers
51 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
33 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 ...
0
votes
0answers
16 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
2answers
24 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
1answer
108 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 ...
2
votes
3answers
39 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 ...
2
votes
2answers
29 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 ...
5
votes
1answer
92 views
+150

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 $K$ ...
0
votes
1answer
28 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
37 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
1answer
142 views

Theorem about equivalent norms.

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms on a normed field. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to ...
1
vote
0answers
41 views

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

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
9 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
43 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
votes
0answers
22 views

TO Find galois group of cubic pooynomial [on hold]

$\text {prove that galois group of }x^3-4x+1\ \text{is}\ S_{3}$?? i have no idea how to approach this problem .please help
1
vote
1answer
44 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 ...
1
vote
3answers
183 views

clarification of algebraic closure and algebraically closed field

Definition of Algebraic closure: An extension $K$ of $F$ is called an algebraic closure of $F$ if (a) $F \subset K$ is algebraic; (b) $K$ is algebraically closed. Given the above definition, I have ...
2
votes
1answer
26 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, ...
0
votes
1answer
25 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
votes
0answers
38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
1
vote
2answers
28 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 ...
4
votes
2answers
88 views
0
votes
2answers
359 views

Why this polynomial is irreducible? [on hold]

Let $K=\mathbb{Z}_p(t)$. How to prove $f(x)=x^p-t$ is irreducible in $K[x]$?
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 ...
1
vote
4answers
1k views

n-th roots of unity form a cyclic group in a field of characteristic p if gcd(n,p) = 1

Let $n$ be a positive integer, and let $\mathbb F$ be a field of positive characteristic $p$ with $\gcd(n,p) = 1$. Where can I find some proofs that the group of all $n$-th roots of unity (in an ...
0
votes
1answer
31 views

Find the degree of a finite field extension

What is the degree of the extension $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{6}):\mathbb{Q}]$? Is it true that this extension is equal to $\mathbb{Q}(\sqrt{2},\sqrt{3})$?
1
vote
2answers
26 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 ...
1
vote
0answers
39 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 ...
2
votes
2answers
35 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 ...
1
vote
1answer
67 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
0answers
33 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
17 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 ...
4
votes
1answer
65 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
0
votes
1answer
23 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 ...
2
votes
0answers
27 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 ...
0
votes
2answers
25 views

Showing that the powers of a root of the p-th cyclotomic polynomial are distinct roots thereof.

Suppose $p$ is a prime number and $p(x) = x^{p-1} + x^{p-2} + ... + x + 1$ is the p-th cyclotomic polynomial and $\gamma$ is a root thereof, so that $p(\gamma) = 0$. I need to show that the following ...
2
votes
2answers
28 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 ...
0
votes
0answers
16 views

Lattice of a subfield

Just a quick question, I saw a question:"what is the lattice of subfields for $GF(p^{30})$?". What is $p$ in this question? Does it mean prime? And what is the answer? Thanks!
1
vote
1answer
30 views

Degree of the extension field

Aren't we supposed to know the degree of the field extension to solve this problem? Did I miss something?
1
vote
1answer
25 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 ...
0
votes
1answer
36 views

Show that a polynomial is still irreducible in a extension field

I have found this question on the Papantonopoulou's Algebra book: Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ with deg$ f(x) = 15$ and deg$ g(x) = 14$. Let $\alpha$ be a ...
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 ...
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 ...
0
votes
0answers
31 views

Fixed field of an automorphism of $\mathbb{C}$ extended from and automorphism of a subfield.

Let us consider the field isomorphism $$ \tau:\mathbb{Q}(\sqrt{2})\rightarrow\mathbb{Q}(\sqrt{2}), a+b\sqrt{2}\mapsto a-b\sqrt{2}. $$ I have read that any isomorphism of a subfield of $\mathbb{C}$ can ...
70
votes
2answers
8k views

The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
0
votes
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
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 ...