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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-1
votes
2answers
69 views

Does there exist any isomorphism between $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 3]$? [duplicate]

Does there exist any isomorphism between the fields $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 3]$ ?
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 ...
2
votes
2answers
86 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 $...
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}(...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
3
votes
0answers
43 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?
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 ...
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 ...
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 $F$ ...
0
votes
1answer
138 views

Would Euclid be satisfied by the construction of the 17-gon given by Gauss?

In our lecture on Algebra we were given the following exercice: Construct the regular 5-gon using straightedge and compass. (only using elementary geometric reasonig) If you construct the length ...
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.
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 $...
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(\...
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?
1
vote
0answers
15 views

Orders of elements in multiplicative groups of fields with positive characteristic

Suppose that $k$ is a field with positive characteristic $p$. I want to show that for each $x\in k$ the set $\{x^n\colon n\in \mathbb{N}\}$ is finite. My intuition tells me that I should use Fermat's ...
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 ...
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
35 views

Cyclic Galois group of even order and the discriminant

I am stuck on the following problem: Let K be a field of characteristic $\neq 2$ and $f\in K[X]$ a separable irreducible polynomial with roots $\alpha_1,\ldots \alpha_n$ in a splitting field $...
3
votes
0answers
39 views

Isomorphism of transcendental extensions

If $a,b$ are transcendental over $\mathbb{Q}$, then it is known that $\mathbb{Q}(a)$ and $\mathbb{Q}(b)$ are isomorphic. Consider a simple case: suppose $a,b,c$ are transcedental over $\mathbb{Q}$. ...
1
vote
1answer
55 views

Three quick queries about fields.

1) Suppose we have some field $F$ then it is known that the smallest subfield of $F$ called $F_0$ say is given by the intersection of all subfields of $F$. Is the reason for this because every family ...
0
votes
0answers
25 views

Circle and line construction of a compex number $z\in\mathbb C$

Let $C\subseteq\mathbb C$ be the field of constructible complex numbers; that is, it includes only the elements $z\in\mathbb C$ which can be constructed with circles and lines. The field $E\subseteq \...
0
votes
3answers
29 views

Example of a communtative ring with two operations where the identity elements are not distinct?

I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity, ...
2
votes
1answer
20 views

Degree of a field extension for three fields

Given field extension $L/K$ and fields $E_1,E_2$ with $$(1)\ K\subset E_1\subset L,\ [E_1:K]=n_1$$ $$(2)\ K\subset E_2\subset L,\ [E_2:K]=n_2.$$ If $\gcd(n_1,n_2)=1$ then $K=E_1\cap E_2$. Proof: ...
1
vote
1answer
52 views

Matrix with irreducible minimal polynomial gives rise to a field

For a field $K$, $A\in Mat_n(K)$ with minimal polynomial (irreducible) $\mu_A(T)\in K[T]$ with $d=\deg\mu_A(T)$. Let $$E=\left\{\sum_{i=0}^{d-1} a_iA^i: a_i\in K\right\}\subset Mat_n(K).$$ Prove that $...
0
votes
1answer
48 views

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$?

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$? I am trying to understand what are in these two structures. My thought is that, if we look at the derivative of $X^p - t$, we ...
1
vote
3answers
55 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate elements of GF$(256)$: meaning that I would like to know what $\alpha^{32}$ or $\alpha^{200}$ is in polynomial form? I am assuming that the primitive ...
2
votes
0answers
26 views

Proof verification: Show that the Frobenius map is surjective.

I would like to prove the following but I would like someone to check my proof. For an algebraically closed field $K$ with characteristic $p$, the Frobenius map $F(x) = x^p$ is surjective What I ...
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{...
0
votes
0answers
29 views

Workability of linear equation solving methods for different fields?

So far, I have mostly done linear algebra over $\mathbb R$ and $\mathbb{C}$. I know there exist very many methods to solve equation systems for those fields, of which a few are Gaussian Elimination, ...
7
votes
1answer
47 views

A field in which every element (that is not 1 or 0) is a root of -1

Let $\mathbb{F}$ be a field with $char(\mathbb{F}) \neq 2$ such that for every element $q \in \mathbb{F}$ if $q \neq 0$ and $q \neq 1$ then there is a power n such that $q^n = -1$. (E.g. $\mathbb{F}_3$...
4
votes
2answers
50 views

Linking two theorems: on algebraic closure and on minimal splitting field

Consider following two statements from same book of Cohn: Basic Algebra. Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting ...
4
votes
0answers
30 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
1
vote
2answers
39 views

Let $K $be a field and $f \in K[x]$. Then there exists a splitting field for $f$ over $K$

Let $K $ be a field and $f \in K[X]$. Then there exists a splitting field for $f$ over $K$. I don't understand what this means, I think I am interpreting it wrongly. Take $x^2+1 \in \Bbb{Q}[X]$ then ...
0
votes
2answers
88 views

Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
0
votes
1answer
47 views

Existence of proper field extension

I am wondering whether the following statement is true or not? Given any field $F$, there exists a proper field extension $K$ of $F$.
2
votes
2answers
39 views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...