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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15
votes
0answers
189 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
0
votes
1answer
51 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...
2
votes
0answers
29 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
5
votes
1answer
53 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
1
vote
1answer
32 views

extend trace and norm from a number field $K$ to $ K \otimes_{\mathbb Q} \mathbb R $

I read somewhere that we can extend the trace and the norm of a number field $K$ to the commutative algebra $ V=K \otimes_{\mathbb Q} \mathbb R$. Before state exactly my question, let me write the ...
6
votes
2answers
280 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
7
votes
2answers
100 views

why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A classic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only ...
0
votes
1answer
14 views

connection between absolute irreducibility and smooth+geometrically connected

Let $C_1$ and $C_2$ be two smooth, projective, and geometrically connected curves over a field $K$ of characteristic $0$. Assume there are two non-constant branch mapping of $K$-curves, $\phi_1\colon ...
2
votes
2answers
42 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
1
vote
0answers
36 views

Cardinality of the set of all field automorphisms of $\mathbb C$ [duplicate]

Does $\mathbb C$ have infinitely many field automorphisms? Does it have uncountably many field automorphisms?
3
votes
0answers
12 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
2
votes
1answer
18 views

Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
8
votes
1answer
69 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
1
vote
1answer
55 views

Proof that $\sqrt{3} \notin \mathbb{Q}(\theta)$ where $\theta^4-2=0$. [on hold]

This is a problem in Robert Ash's lecture notes in Algebraic Number Theory. I have to prove that $\sqrt{3} \notin K=\mathbb{Q}(\theta)$ where $\theta^4-2=0$, using the fact that ...
0
votes
0answers
9 views

Trace of an element in a separable field extension

Let $L=K(\alpha)$ be a finite separable field extension of $K$ of degree $n$ and let $\alpha$ have minimal polynomial $f(X)\in K[X]$ with roots $\alpha=\alpha_1,...,\alpha_n$. Write ...
2
votes
1answer
28 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
1
vote
1answer
38 views

Is there a way to generate groups, rings, fields, etc.? [on hold]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
9
votes
1answer
237 views

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
0
votes
2answers
65 views

Create a field from set of 2 elements.

Can we always create a field from a set of at least $2$ elements? For addition I considered a function: $A\times A \rightarrow A$. If $a+b=b+a=a+a \rightarrow a $. If $b+b \rightarrow b$. Is it ...
-4
votes
0answers
49 views

Advanced Linear Algebra [on hold]

Let F be any field. Let $f_1$, $f_2$, $f_3$ be the following three polynomials in $F[X]=P(F)$ $$f_1=X+1$$ $$f_2=X^2-1$$ $$f_3=X^2+3X+1 $$ Do $f_1$, $f_2$ and $f_3$ form an $F$-basis for $P_1(F)$, ...
1
vote
0answers
17 views

Computing number of irreducible polynomials of degree n over $\mathbb{F}_q$

When I try to find the number of irreducible polynomials (of degree n) over a finite field I first look for the number of $\alpha \in \mathbb{F}_{q^n}$ such that ...
0
votes
2answers
37 views

Showing that any field extension of a finite field is simple

We know that the multiplicative subgroup $F^\times$ of a finite field $F$ is cyclic. Use this to show that any field extension of a finite field is simple. Any clues?
-1
votes
1answer
19 views

I have to show it is isomorphic to $K = GF(p^{kd})$ [on hold]

Suppose $F = GF(p^k)$ is a finite field. I know $F[C]$ is a field extension of $F$ with degree $d = \deg m$, and I have to show it is isomorphic to $K = GF(p^{kd})$ (where $C$ is a companion matrix ...
1
vote
1answer
83 views

$f(x) \mid g(x) \iff g(x) \in \langle f(x) \rangle$. Isn't this trivial?

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
2
votes
0answers
28 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...
1
vote
3answers
25 views

Finding an isomorphism between polyomial quotient rings

Let $F_1 = \mathbb{Z}_5[x]/(x^2+x+1)$ and $F_2 = \mathbb{Z}_5[x]/(x^2+3)$. Note neither $x^2+x+1$ nor $x^2+3$ has a root in $\mathbb{Z}_5$, so that each of the above are fields of order 25, and hence ...
36
votes
3answers
9k views

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
2
votes
1answer
49 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
2
votes
1answer
49 views

Galois group and traslations by rational numbers.

Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$. Now my question: ...
1
vote
1answer
23 views

Isomorphism of quadratic extensions (of a number field)

I think we agree that two (squarefree) quadratic extensions of $\mathbb Q$, say $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ are not isomorphic. Now consider the following tower of fields ...
1
vote
2answers
1k views

When the group of automorphisms of an extension of fields acts transitively

Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ ...
1
vote
0answers
22 views

$E/F$ is a normal extension iff $E$ is a splitting field for some polynomial $f\in F[X]$.

$E/F$ is a finite extension. Prove that $E/F$ is a normal extension $\iff$ $E$ is a splitting field for some polynomial $f\in F[X]$. An extension $E/F$ is called normal if it is algebraic and ...
1
vote
2answers
39 views

Which of these statements about the field extension $\mathbb{R}/\mathbb{Q}$ are true?

We know that $\mathbb R$ is an extension of $\mathbb Q$. Justify the following (true /false): $[\mathbb R:\mathbb Q]<\infty$ $[\mathbb R:\mathbb Q]=$ countably infinite / uncountably ...
0
votes
2answers
42 views

Relationship between cardinalities of $K$ and $F$, if $K/F$ is an algebraic extension [closed]

Let $F$ be a field. If $K$ be an algebraic extension of $F$ what can be concluded about the following: If $|F|$ is finite what will $|K|$ be? If $|F|$ is infinite what will $|K|$be ...
4
votes
2answers
42 views

Degree of minimal polynomial

The minimal polynomial of $a$ over $\mathbb{Q}$ is quadratic. The minimal polynomial of $b$ over $\mathbb{Q}$ is cubic. Is the minimal polynomial of $a+b$ necessarily of degree $6$? If so, what is ...
0
votes
0answers
32 views

Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ .

Let $E/F$ be a finite extension and it is a Galois extension. Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ . I cant start at all.How should I begin?
0
votes
1answer
30 views

Why the complex number system is not an ordered field [duplicate]

In high school, we are taught that we do not have $2i < 3i$, i.e., the complex number system is not an ordered field. (Real number, for example, is an ordered field. For example, $2 < 3$). ...
1
vote
1answer
81 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
0
votes
1answer
20 views

Equivalent Definitions of Prime Subfield

I found two definitions for a prime subfield $K$ of a field $F$. 1. Wolfram $-$ $K$ is the subfield of $F$ generated by the multiplicative identity $1$ of $F$. 2. ProofWiki $-$ $K$ is the ...
1
vote
2answers
59 views

Prove that field $Q(x)$ is a field of fractions of ring $F[x]$

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. How can I prove that field $Q(x)$ is a field of fractions of ring $F[x]$? And also why is it that field $Q((x))$ ...
6
votes
4answers
101 views

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$

I've got that $[\mathbb{Q}(\sqrt{5}+\sqrt[3]{2}):\mathbb{Q}] \in \{1,2,3,6\}$ because it's going to divide $[\mathbb{Q}(\sqrt{5},\sqrt[3]{2}):\mathbb{Q}]=6$. Clearly it is not $1$. I want to show that ...
2
votes
2answers
66 views

Example of two field extensions such that their tensor product is not a field

Example of two fields $K$ and $L$, which are extensions over $k$, such that $K\otimes_k L$ is not a field. Here is what I did. But I am a little bit unsure. Can someone suggest anything, or perhaps ...
1
vote
1answer
41 views

Why is is $K(\alpha,\beta)/K(\alpha)$ algebraic if $K(\alpha,\beta)/K(\beta)$ is algebraic? [duplicate]

Let $K$ be a field, and let $\alpha$ be transcendental over $K$ and algebraic over $K(\beta)$. We have a Hasse diagram of field extensions Now, by reduction to absurdity $\beta$ must be ...
1
vote
1answer
39 views

Is the Galois correspondence still valid if automorphisms aren't required to be the identity on the base field?

Let $E$ be a finite extension of $F$, and $\mathrm{Aut}(E|F)$ be the group of those automorphisms of $E$, which take $F$ to itself (but not necessarily identity on $F$. Let $X$ be the collection of ...
2
votes
1answer
89 views

Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
1
vote
1answer
46 views

Every finite abelian extension of Q contains a totally real subfield of index 2?

I can reduce this to the case of cyclotomic field extensions, by embedding the abelian extension into a cyclotomic extension and using the "sliding-up" lemma. I am stuck on how to prove this for the ...
3
votes
0answers
41 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 ...
-1
votes
1answer
45 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
1
vote
0answers
57 views

Quotient field - base change

For my master thesis, I need to examine the following statement: $Frac(R) \otimes_{k} L \cong Frac(R \otimes_{k} L)$, where $R$ is an integral domain over the perfect field $k$ and $L$ is a finite ...
1
vote
3answers
81 views

$\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$

Show that $\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$, but is it more? Are these fields equal? $\mathbb{Q}(\sqrt2)=\{a+b\sqrt2 |a,b \in \mathbb{Q}\}$ ...