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 topic....

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2
votes
1answer
58 views

Solvability and reducibility of a polynomial in a “chain” of finite fields

This question is generalized based on my previous question: Is $x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$? Problem: Consider an irreducible polynomial $f = x^4 + x^3 + 1$ in ...
2
votes
1answer
63 views

What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
0
votes
0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
2
votes
2answers
42 views

Is there an infinite field F with char(F)=p and not algebraically closed field?

Is there an infinite field F with characteristic of the field $F$ is $p$ (p is prime) and not algebraically closed field ?
3
votes
3answers
283 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
4
votes
1answer
51 views

Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...
11
votes
0answers
70 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
8
votes
3answers
109 views

Non-trivial example of algebraically closed fields

I'm beginning an introductory course on Galois Theory and we've just started to talk about algebraic closed fields and extensions. The typical example of algebraically closed fields is $\mathbb{C}$ ...
1
vote
1answer
50 views

Ring->module->$R$-algebra, Field->Vectorspace->algebra

I haven't done any mathematics for a long time, and I have forgotten some things. I want to try to remember some of the words and how they interact. A module is a 'vectorspace over a ring' rather ...
1
vote
3answers
55 views

Integral closure of Gaussian Integers

I am considering $\mathbb{Z}[i]\subset\mathbb{Q}(i)$ Now I have a short note here that says that there are elements of $\mathbb{Q}(i)$ which are not integral over $\mathbb{Z}[i]$ 'because $\mathbb{Q}(...
3
votes
1answer
26 views

What does a finite extension look like?

Let $F$ be a field. Let $\Gamma$ be an indexing set. Let $\lbrace \theta_i\rbrace_{i\in\Gamma}$ be a collection of of elements of $\overline{F}$. Let $L=F(\theta_i\text{ }|\text{ }i\in\Gamma)$ (the ...
1
vote
2answers
1k views

Proof: Zero is less than one

How can one show/prove that $0<1$? $1$ is not actually defined to be greater than zero, but I think it can be proven. I already know that the real numbers are an ordered field and I am familiar ...
0
votes
0answers
24 views

$F(X)$ as a subfield of $F((X))$ of formal Laurent series

$F(X)$ is a subfield of $F((X))$ by considering the Laurent expansion of rational functions at the origin. So what is indeed the degree of this field extension $F((X))/F(X)$? Or this is an infinite ...
1
vote
0answers
71 views

Why is analysis over the complex numbers so useful vs say other fields?

First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime ...
5
votes
1answer
221 views

A question concerning non-algebraic extension.

Let $\tau:F \to \overline{F}$ be a field embedding. Then is $\overline{F}/\tau(F)$ algebraic extension? I don't think so but I cannot find a counterexample. Would you let me know a counterexample?
-1
votes
2answers
69 views
3
votes
5answers
101 views

Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$?

Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$? I know that $[\mathbb{Q}(\sqrt{2},e^{2i\pi/3}):\mathbb{Q}]=2\times2=4$. By using WolframAlpha (cheating), I know ...
1
vote
3answers
63 views

prove that homomorphism (rings) from a field to ring is bijective or the zero homomorphism.

$F$ is a field and $R$ is a ring.$\:\phi :F\rightarrow R$ is a ring homomorphism. I need to prove that it is bijective or it is $\phi =0$. I tried to use some how the fact that I have opposites in F, ...
3
votes
2answers
102 views

In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
2
votes
2answers
87 views

Show that $K(a_1, \dots , a_n)=K[a_1, \dots , a_n]$

Let $L/K$ be a field extension and $a_1, \dots a_n\in L$, such that $a_1$ is algebraic over $K$, $a_2$ is algebraic over $K(a_1)$ and in general, $a_i$ is algebraic over $K(a_1, \dots , a_{i-1})$ for $...
2
votes
0answers
164 views

Separability of field extensions

I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra. $1)$ $L/k$ is ...
2
votes
2answers
41 views

Show that $[L:K]=1 \Leftrightarrow L=K$

Let $L/K$ be a field extension. I want to show that $$[L:K]=1 \Leftrightarrow L=K$$ $$$$ I have done the following: For the direction $\Rightarrow \ : $ Since $[L:K]=1=\text{dim}_KL$ we ...
0
votes
0answers
34 views

Show that $n$ is a divisor of $[L:K]$

Let $L/K$ be a field extension. I want to show that if the extension $L/K$ is finite and $a\in L$ has a minimal polynomial of degree $n$, then $n$ is a divisor of $[L:K]$. $$$$ I have done the ...
1
vote
1answer
30 views

Determining the minimal polynomial of $\omega := e^{2πi/p}$ over $\mathbb{Q}[\omega + \omega^{-1}]$

Let $p ≠ 2$ be a prime number, and $\omega = e^{2πi/p}$. I now want to find the minimal polynomial of $\omega$ over the field $\mathbb{Q}[\omega + \omega^{-1}]$. I must admit that I don't really know ...
1
vote
0answers
23 views

Radical-solvable extensions

I'm studying Field and Galois Theory with different books and now I have a doubt about what is the exact statement of Galois' theorem. Some books define radical and solvable extensions but other books ...
7
votes
1answer
185 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...
0
votes
0answers
23 views

Cyclotomic polynomials being irreducible over Q

So, task is to, using algebra, write polynomial $X^n-1$ as a product of irreducible polynomials over Q. Our prof told us that the solution is : $X^n-1 = \prod_{d|n} \Phi_d(x)$ where $\Phi_d(x)$ is d-...
2
votes
0answers
25 views

Why working in compact spaces?

I am trying to study moduli spaces of stable curves with n-marked points, $M_{0,n}$. However, in general the texts generally talk about the closure of this space, $\overline{M_{0,n}}$. My question, ...
1
vote
1answer
68 views

Finite Field Question: Which of the followings are true?

I have the following True or False question that I am having trouble getting it correct. I've written down my thoughts on each choice. If anyone could verify my thoughts or tell me where I made a ...
3
votes
2answers
36 views

$X^3-2$ splits completely in an extension of $\mathbb F_7$

My question concerns the following problem: Let $K=\mathbb F_7[T]/(T^3-2)$. Show that $X^3-2$ splits into linear factors in $K[X]$. Write $K\simeq \mathbb F_7[\alpha]$ for a root $\alpha\in \...
0
votes
1answer
15 views

Extending isomorphism to compositum of fields

Let $F$ be a field and $\Gamma$ be an indexing set (possibly infinite). Let $K$ be another field. There is an isomorphism $\sigma:F\longrightarrow K$. Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a ...
0
votes
4answers
45 views

For an algebraically closed field $k$, an ideal $I$ of $k[x]$ is maximal if and only if $I = (x-c)$

This is an exercise $4.21$ on a page $155$ from a textbook "Algebra: Chapter $0$" by P.Aluffi. Let $k$ be an algebraically cloased field, and let $I \subseteq k[x]$ be an ideal. Prove that $I$ is ...
3
votes
0answers
44 views

Isomorphism of fields

In set-theory, one of the standard result (Bernstein's theorem) is that if there is an injection from $A$ to $B$ and an injection from $B$ to $A$, then there is a bijection from $A$ to $B$. Consider ...
1
vote
0answers
68 views

How to determine redundant elliptic curves?

When we enumerate elliptic curves $y^2 = x^3 + ax + b$ over a finite field, how do we determine redundant ones, i.e. ones that are equivalent to others?
0
votes
1answer
18 views

Compositum of an infinite family of fields

Let $F$ be a field and $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of subfields of $\overline{F}$. Let $E\subseteq \overline{F}$ be the ...
0
votes
2answers
19 views

is the compositum of a family of algebraic extensions algebraic?

Let $F$ be a field contained inside another field $K$. Let $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of algebraic extensions of $F$ ...
4
votes
2answers
93 views

Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
3
votes
3answers
489 views

Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
1
vote
2answers
55 views

Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
2
votes
2answers
43 views

p-adic distances

We take $\mathbb{Q}_p$ to be the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$. If $x=\sum_{j=k}^{\infty} a_jp^j$ is some element in $\mathbb{Q}_p$, then how exactly does $|\cdot|_p$ extend? ...
2
votes
0answers
43 views

Points on a p-adic circle

I was wondering if anybody could point me to any interesting geometry (if there is any) that the p-adic circle has. Specifically, let $G_p=\{ (x,y)\in (\mathbb{Q}_p)^2 \,:\, x^2+y^2=1\}.$ Does $G_p$ ...
7
votes
1answer
124 views

Galois group of $x^5-5x+10$

I was illustrating the theorem on solvability by radicals through some examples of degree $5$ polynomials. One I chose was $x^5-5x+10$. I was (perhaps wrongly) going to prove that the Galos group is $...
2
votes
1answer
60 views

Galois group of function field

Let $K$ be an arbitrary field, and $K(t)$ denote the field of rational functions in $t$, i.e. function field on $K$. If $K$ is algebraically closed field, then $\mathrm{Gal}(K(t),K)\cong \mathrm{...
1
vote
1answer
87 views

Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
0
votes
0answers
43 views

$i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$

$i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$ where operations in proper superset/field maps to a proper subfield. Is this of mapping between superfields to subfields of any ...
1
vote
1answer
31 views

Showing $K(\alpha^2) = K(\alpha)$ for some field $K$ with $[K(\alpha) : K] = p$

Let $K, L$ be fields, $K \subseteq L$ and $[K(\alpha) : K] = p$ for a prime number $p ≠ 2$, and some $\alpha \in L \backslash K$ that is algebraic over $L$. I now want to show that $K(\alpha^2) = K(\...
5
votes
2answers
311 views

Elements in finite field extensions

Let $A,K$ be finite fields with $K\supset A$. If $[K:A]=3$, I would like clarification as to why, if $x$ is not a square in $A$, then $x$ is not a square in $K$. My notes just mention this fact, but ...
5
votes
1answer
97 views

Degree of the difference of two roots

Let $f\in \mathbb Q[x]$ be an irreducible monic polynomial of degree $n$ and let $\alpha,\beta\in \overline{\mathbb Q}$ be two distinct roots of $f$. Is it possible to find a lower bound on the degree ...
4
votes
1answer
45 views

An extension of $\mathbb{Q}$ which contains the $n$-th roots of every element

Consider $\mathbb{Q}$, the field of rational numbers. Let $K_1\subseteq \mathbb{C}$ be the (minimal) splitting field of the family $\{x^n-a\colon a\in\mathbb{Q}, n\geq 1\}$. Let $K_2\subseteq \...
0
votes
0answers
25 views

Fields that are vector-spaces over the set of real numbers [duplicate]

Can somebody enlighten me on how to prove that there exists no field that's also a vector space over the real numbers of dimension greater than 2?