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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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 ...
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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 $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
3
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1answer
57 views

Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
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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 ...
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0answers
34 views

What is the possible number (supremum) of subfields of $\mathbb{F}$?

Let $\mathbb{F}$ be field. it is a finite dimensional extension over $\mathbb{Q}$. So let $B=\{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb{F}$ over $\mathbb{Q}$. From the finite dimension ...
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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.
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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 ...
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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 ...
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 ...
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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 ...
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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 $x$} & ...
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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 ...
7
votes
1answer
76 views

If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.

Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + ...
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2answers
58 views

Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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1answer
64 views

What are the groups $\text{Hom}(F^\times\!, F^+\!)$ and $\text{Hom}(F^+\!, F^\times\!)$?

Background. Exercise 36 in Rose's A Course On Group Theory reads Prove that there is no field $F$ with $F^\times \cong F^+$. The problem is solved in characteristic $\ne 2$ by considering $-1$ ...
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1answer
93 views

Query on an example of Morandi's Field and Galois Theory, regarding the degree of a field extension.

I am going through Morandi's Field and Galois Theory, and I am looking at Example $1.5$, Chapter I. It says (more or less): If $k$ is a field, let $K=k(t)$ be the field of rational functions in ...
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2answers
2k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
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0answers
38 views

How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
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1answer
43 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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2answers
56 views

How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
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0answers
31 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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3answers
99 views

$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in ...
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1answer
12 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
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2answers
27 views

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$.

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$. **My try **: $x^3-2=(x-2^\frac{1}{3})(x^2+2^\frac{1}{3}x+2^\frac{2}{3})$ On solving I am getting the roots as ...
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1answer
38 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
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0answers
21 views

Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
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1answer
29 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L ...
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0answers
32 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
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0answers
38 views

What does this theorem mean, exactly?

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $F_p[X]$ of degree $d$ where $d$ runs through all the divisors of $n$. I don't even get the ...
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0answers
8 views

Question on separable degree

I asked a similar question before but I didn't get a satifying answer, so I'm posting it again. Let me first define terms: Def1 Let $E/F$ be an algebraic field extension and $\bar F$ be an ...
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1answer
80 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
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2answers
379 views

Shortest irreducible polynomials over $\Bbb F_p$ of degree $n$

For any prime $p$, one can realize any finite field $\Bbb F_{p^n}$ as the quotient of the ring $\Bbb F_p[X]$ by the maximal ideal generated by an irreducible polynomial $f$ of degree $n$. By dividing ...
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1answer
24 views

Finding proper subfields

Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly. Two of ...
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votes
5answers
70 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
0
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1answer
33 views

Relation between $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$

I have the following exercise in my homework: Are $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$ isomorphic, where $w = \textrm{cis}((2\pi)/3)$? Prove your answer. I think they are, but I'm ...
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1answer
41 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
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1answer
47 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
0
votes
1answer
31 views

Is the statement of the theorem correct?

I have been asked to prove this:: $f,g$ are polynomials over a field $F$ .Prove that if $f,g$ are relatively prime then $f,g$ have no common roots in any extension of $F$. But I wonder why is ...
4
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2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
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1answer
45 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...
7
votes
1answer
51 views

On the existence of field morphisms

Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
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4answers
35 views

Basis of $\mathbb{Q}[\sqrt[3]{2}]$

How do I prove that $1, \sqrt[3]{2}, (\sqrt[3]{2})^2$ is a basis of $\mathbb{Q}[\sqrt[3]{2}] = \{ a + b \sqrt[3]{2} + c (\sqrt[3]{2})^2\: a,b,c \in \mathbb{Q} \}$. It's one of these cases where the ...
2
votes
2answers
39 views

Injectivity and norm function on finite fields [closed]

Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by $$f(x)=\alpha x^q+\alpha^q x$$ for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 ...
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votes
1answer
33 views

Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
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2answers
23 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
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3answers
95 views

How do I prove this field homomorphism is an isomorphism?

The question is as follows. Let $F$ be a finite field with unit $1$ not equal to zero. Let the function $f: F \to F$ be given by $f(x) = x^3$, where the $\operatorname{char}(F) = 3$. Prove it is a ...
4
votes
1answer
49 views

Is the primitve element of $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ always $\alpha_1 + \alpha_2 + \cdots$?

I have dealt with a number of algebraic field extensions $\mathbb{Q}[\alpha_1, \alpha_2, \ldots]/\mathbb{Q}$ and the primitive element was always $\alpha_1 + \alpha_2 + \cdots$. Is this generally true ...
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votes
0answers
12 views

Is $K[X]/fK[X]$ = $K[X]/f$?

Let K be field. f is polynomial in K[X]. Is it the same: $K[X]/fK[X]=K[X]/(f)$ ? In other words are the elements equal h+fK[X]=h+f ? ($h \in K[X]$
6
votes
3answers
85 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
4
votes
1answer
73 views

Finite fields and their subfields

Let $\mathtt{F}$ and $\mathtt{F'}$ be two finite fields of order $q$ and $q'$ respectively. Then: $\mathtt{F'}$ contains a subfield isomorphic to $\mathtt{F}$ if and only if $q\le q'$ ...