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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8
votes
1answer
172 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
0
votes
2answers
42 views

How to prove that there exists a countable subfield of real numbers which is mapped into itself for any function f?

Given a function $f: \mathbb{R} \rightarrow \mathbb{R}$, how should I go about proving that there exists a countable subfield of $\mathbb{R}$, say $K$, which is mapped into itself? i.e., $f(K)\subset ...
4
votes
1answer
35 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
2
votes
3answers
39 views

Computing basis of a field extension

Suppose we have an irreducible polynomial $p(x)$ of degree $n$ in $F[x]$, where $F$ is a field. Let $K$ be the field $F[x] / (p(x))$. We can consider the extension of $K$ over $F$ as a vector space ...
5
votes
0answers
126 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
0
votes
1answer
21 views

irreducibility of $x^2-a$ in $\mathbb{Z}_2[a]$

Let $a$ be transcendental over $\mathbb{Z}_2$ and let $F=\mathbb{Z}_2(a)$. Prove $p(x)=x^2-a$ is irreducible over $F$ I've been trying to understand this for a while now, but I'm ...
0
votes
0answers
22 views

$k(u)=k(t)\Leftrightarrow ht(u)=1$

In my lecture we proved the following statement: $u\in k(t)\backslash k$ show $k(u)=k(t)\Leftrightarrow ht(u)=1$ But I don't understand the proof we did (I'll put little numbers over the parts I ...
0
votes
1answer
25 views

Showing a polynomial is irreducible over an extension field.

Show that the polynomial $$ x^3 - 3$$ Is irreducible over $$ Q(i, \sqrt2 ) $$ I'm a little stuck as I don't think I can use Eisenstein's criterion as we're not over the rationals. Also I know ...
3
votes
1answer
24 views

A counterexample about an inequality- Field extensions

Consider $A$ and $B$ two intermediate fields of the field extension F/K. I have already proved that $[AB:K]\leq {[A:K][B:K]\over [A\cap B: K]}$. I would like to find a simple example (for example, ...
0
votes
1answer
41 views

multiplicative group of infinite fields

I am stuck with the proof of below expression; "If F is an infinite field, then no infinite subgroup of F* (the multiplicative group of F) is cyclic." anyone can help?
0
votes
2answers
44 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
1
vote
1answer
67 views

Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...
1
vote
1answer
32 views

$\mathbb{F}_p[X]/(X^2+X+1)$ field iff $p \equiv 2 $ mod $ 3$

Let $p$ be prime. Prove $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 $ mod $ 3$. So: If $p \equiv 2 $ mod $ 3$, I have to show that every element of $\mathbb{F}_p[X]/(X^2+X+1)$ has ...
2
votes
3answers
55 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
5
votes
2answers
70 views

Let $F$ be a field of order $2^n$. Prove that characteristic of $F$ is 2.

I figure that Lagrange's theorem and the fact that the characteristic of an integral domain is either $0$ or prime should be used, but just can't figure it out exactly.
2
votes
1answer
97 views

How to show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$

So as title says I wanna show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ So I know that the minimal polynomial of $\mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ is $x^{4}-6x^{2}+7$ ...
1
vote
1answer
46 views

Question about construction of an algebraic closure

$A=K[x]$,$\mathfrak{m}$ is a maximal ideal containing a principle ideal of $A$. every element of $K[x]/\mathfrak{m}$ can be described by form $f+\mathfrak{m}$. every element of $K$ is the ...
1
vote
1answer
58 views

$\exists a \in \mathbb{F}_{11}$ such that $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field

Prove that there exists an element $a \in \mathbb{F}_{11}$ such that the quotient ring $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field. I wrote that it is equivalent to showing that there is ...
0
votes
0answers
47 views

Galois group of a particular polynomial

What is the Galois group of the polynomial $X^n − 3$ over $\mathbb Q$? (Here $n$ is greater than $2$.)
0
votes
1answer
56 views

Question about construction of an algebraic closure of a field

In Constructing algebraic closures by Keith Conrad, the author writes: Let $K$ be a field. We want to construct an algebraic closure of $K$, i.e., an algebraic extension of $K$ which is ...
1
vote
2answers
70 views

Prove that $-1 \cdot x=-x$ [duplicate]

While working on a proof for class, I came to a point where I couldn't go any further without knowing that $-1 \cdot x=-x$. Is there a way to prove this using the axioms of a field?
-2
votes
2answers
64 views

{0} is unique maximal ideal when F is field [duplicate]

Let $R$ be a ring. Show that R is a field if and only if $\{ 0 \}$ is the unique maximal ideal of $R$. Thank you
2
votes
2answers
53 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
2
votes
1answer
47 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
0
votes
2answers
28 views

$k$ algebraically closed field $\Rightarrow$ $V(f)\subset k^2$ infinite

Let $k$ be an algebraically closed field, and $f\in k[X,Y]$ a non-constant polynomial. Show that $V(f)\subset k^2$ is infinite. We solved this exercise in my tutorial class, but I have some questions ...
2
votes
1answer
72 views

If $F\subseteq\mathrm{Mat}_n(\mathbb{Q})$, then $[F:\mathbb Q]\leq n$?

Let $F$ be a field contained in the ring of $n\times n$ matrices over $\mathbb Q$. Prove that $[F:\mathbb Q]\leq n$. I have an idea to consider a degree $n$ extension $K$ of $\mathbb Q$ and left ...
1
vote
0answers
37 views

Composite of two simple extensions

Let $a, b$ be algebraic elements over a field $K$ and suppose at least one of these two elements is separable over $K$. Then, prove that there exists $c$ such that $K(a, b) = K(c)$.
3
votes
1answer
51 views

Subfield of $\mathbb R$ with Algebraic closure as $\mathbb C$.

Does there exist a proper subfield of $\mathbb{R}$ whose algebraic closure is $\mathbb{C}$ ? A weaker question: Does there exist a proper subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is ...
1
vote
1answer
41 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
2
votes
1answer
39 views

What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I ...
3
votes
1answer
52 views

Equivalent definition of purely inseparable field extension concerning extensions of morphisms.

Suppose $F$ is a field of characteristic $p$. I know there are many equivalent defintions that a field extension $K/F$ be purely inseparable, e.g., every separable element in $K$ over $F$ is in $F$, ...
12
votes
3answers
505 views

Characterization of a subfield $K \varsubsetneq \mathbb {C}$ and $x\in \mathbb{R}$

Characterize $x \in \mathbb R$ such that : There exist a subfield $K \varsubsetneq \mathbb C$ such that $K(x) = \mathbb C$ -All subfields $K$ of $\mathbb{C}$ contain $\mathbb Q$, then all $x\notin ...
8
votes
2answers
97 views

Can we make $\mathbb{Z}$ into a field?

This is probably an elementary question about fields, but I think it is a little tricky. Can we make the integers $\mathbb{Z}$ into a field? Let me be more precise. Is it possible to make ...
0
votes
2answers
37 views

Dimension of $K\subset L(\alpha)$ where $L$ is a field extension of $K$

Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a ...
2
votes
0answers
38 views

Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
1
vote
1answer
41 views

Is $f/\gcd(f,f')$ always separable?

Suppose $f$ is a nonzero polynomial over an arbitrary field. If $g=\gcd(f,f')$, is it true that $f/g$ is always separable? I was trying to show $\gcd(f/g,(f/g)')=1$. If $d$ is a common divisor of ...
3
votes
1answer
33 views

A polynomial with solvable Galois group and solution by radicals [duplicate]

Suppose $f(x)\in \mathbb{Q}[x]$ has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of $f(x)$ in terms of ...
0
votes
0answers
47 views

What is an algebraic expression over a field structure?

I am working on a problem, and I am not understanding the language very well. Here is the setup of the problem: Consider the set $\{ 0, 1, 2 \}$ with the operations addition $(+)$ modulo $3$ and ...
0
votes
6answers
42 views

Prove field extension is a field

I have a field extension $\mathbb Q (2^{1/3}) = a + b2^{1/3} + c2^{2/3}$ where $a,b,c\in \mathbb Q$. I want an elementary proof it indeed is a field. How to go about proving it contains its inverse ...
0
votes
2answers
61 views

Prove that $\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=2^{n}$ [duplicate]

Let $p_{1},p_{2},\ldots,p_{n}$ be $n$ primes,$\left(p_{i},p_{j}\right)=1$ if $i\neq j$ . Prove that ...
0
votes
0answers
37 views

Prime polynomials over GF(q)

Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
1
vote
1answer
45 views

Is there a way to classify the finite fields where $x^2+1$ has a root?

It is a well-known theorem in number theory that $-1$ is a square in $\mathbb{F}_p$ if $p\equiv 1\pmod{4}$, and $-1$ is not a square if $p\equiv 3\pmod{4}$. Furthermore, $-1$ is easily seen to be a ...
1
vote
2answers
41 views

Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
0
votes
1answer
30 views

Find the fixed field of $f\colon K(X)\to K(X)$ given by $f(X)=1/X$

Let $K(X)$ denote the field of fractions of the polynomial ring $K[X]$ over a field $K$. Find the fixed field of the automorphism $f\colon K(X)\to K(X)$ given by $f(X)=1/X$.
0
votes
1answer
54 views

Why principal ideal should be commutative?

According to the definition of Principal Ideal it should be commutative. What if the ring is not commutative? Which means $ar\neq ra$ where $a\in I, r \in R$. Does it lead to a contradiction? Because ...
2
votes
1answer
95 views

If $f$ is an irreducible rational polynomial then all the roots over $\mathbb{C}$ are distinct

I'm trying to show that if $f \in \mathbb{Q}[t]$ is irreducible then all the roots of $f$ in $\mathbb{C}$ are distinct. My first issue, am I right in thinking that the roots are distinct iff ...
3
votes
0answers
44 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 ...
0
votes
1answer
57 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
4
votes
0answers
41 views

Nature of a particular extension [closed]

Let $K$ be a field, and suppose that $f$ is an automorphism of $K$ which has infinite order. Let $F$ be the fixed field of $f$. If the extension of $K/F$ is algebraic, show that $K$ is normal over ...
0
votes
1answer
33 views

Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...