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 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
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1answer
17 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 ...
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0answers
6 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
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1answer
33 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 ...
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1answer
56 views

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 ...
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2answers
61 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 ...
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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 $P_1(F)$, ...
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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?
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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
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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 ...
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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
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1answer
47 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
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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 ...
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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 ...
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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 ...
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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?
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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, $2 < 3$). ...
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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 ...
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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
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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 $Q((x))$ ...
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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 ...
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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? $\mathbb{Q}(\sqrt2)=\{a+b\sqrt2 |a,b \in \mathbb{Q}\}$ ...
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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
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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 ...
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1answer
61 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 ...
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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 ...
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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
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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.
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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 ...
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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 ...
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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
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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 ...
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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
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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
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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
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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 ...
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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 $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
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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 $x$} & ...
1
vote
1answer
37 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...