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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2
votes
1answer
50 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
6 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
22 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 ...
0
votes
0answers
7 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 ...
1
vote
1answer
34 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 ...
8
votes
1answer
65 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 ...
0
votes
2answers
63 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 $P_1(F)$, ...
0
votes
0answers
16 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
34 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 ...
2
votes
0answers
27 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 ...
2
votes
1answer
48 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
22 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 ...
0
votes
2answers
42 views

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

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 ...
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 ...
4
votes
2answers
41 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
27 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?
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 ...
0
votes
1answer
28 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, $2 < 3$). ...
0
votes
1answer
19 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 ...
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
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 ...
6
votes
4answers
99 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 ...
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
2answers
57 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 $Q((x))$ ...
1
vote
0answers
55 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? $\mathbb{Q}(\sqrt2)=\{a+b\sqrt2 |a,b \in \mathbb{Q}\}$ ...
2
votes
0answers
25 views

Exercise about cyclic field extension

I am having hard time to solve following exercise. Let $\Omega$ be the algebraic closure of a field $k$. a) Suppose that every finite extension of $k$ is cyclic. Prove that it exists $\sigma \in ...
2
votes
1answer
35 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
1
vote
1answer
62 views
+50

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 ...
2
votes
1answer
21 views

L|K field extension and $deg(f)\nmid[L:K]$

Let L|K be finite field extension and $f \in K[X]$ is irreducible with $deg(f) > 1$. Show that, if $f\nmid[L:K]$ then f has no zeros in L. Is it true? For ex. $f=x^3+x$ and $[Q(\sqrt 5,i):Q]$. f ...
3
votes
2answers
86 views

Polynomials over a finite field

Let $\mathbb{F}_p$ be a finite field where $p$ is a prime. Consider the following set of polynomials over $\mathbb{F}_p$: $$G_n(p)=\{{x+a_2x^2+\cdots+a_nx^n\mid a_i\in \mathbb{F}_p}\}.$$ Is ...
0
votes
1answer
66 views

Galois Group of $x^4 - x^2 - 3$

Find the Galois Group of $x^4 - x^2 - 3$ This is a qual question. I don't know how to find the splitting field of this polynomial.
1
vote
2answers
46 views

Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
1
vote
1answer
45 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
2
votes
1answer
25 views

Finding the subfields of the cyclotomic field of order $5$

This is part of an exercise from Hungerford's Algebra: Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$. ...
3
votes
1answer
19 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
0
votes
0answers
30 views

Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes 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 ...
4
votes
0answers
53 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
1
vote
1answer
49 views

Is the finite union of algebraic curves an algebraic curve? [closed]

Is the finite union of algebraic curves an algebraic curve? I'm kind of new to the study of algebraic curves and I believe this is intuitive.
2
votes
1answer
34 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 ...
2
votes
2answers
53 views

Naïve groups, fields and ideals

Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right ...
4
votes
2answers
57 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
1
vote
1answer
38 views

Reducing splitting field

If we have splitting field: $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{2-i\sqrt{6}}) $$ we can multiply these two zeroes and get $\sqrt{10}$ so we have $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
1
vote
2answers
47 views

Are the fields $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ and $\mathbb{Q}(\sqrt[7]{16})$ equal?

I have trouble with these field extensions. Is field $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ equal to field $\mathbb{Q}(\sqrt[7]{16})$? We can $\sqrt[7]{16}+3 \sqrt[7]{8}$ express as ...
2
votes
1answer
31 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
1
vote
3answers
42 views

List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & ...