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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5
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1answer
76 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 ...
1
vote
1answer
29 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
24 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?
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0answers
13 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 ...
2
votes
0answers
22 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 ...
4
votes
1answer
28 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 $...
2
votes
0answers
36 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
3answers
48 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 ...
1
vote
1answer
47 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 $...
1
vote
1answer
54 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
24 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
19 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: ...
0
votes
1answer
43 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 ...
2
votes
1answer
41 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{...
2
votes
1answer
734 views

Minimal polynomial and field extension

Suppose the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$. Does the above imply ...
2
votes
0answers
23 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 ...
0
votes
0answers
24 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, ...
8
votes
1answer
129 views

$a$ transcendental $\implies a^a$ is transcendental?

Suppose $a\in \mathbb{C}$ is not a algebraic number. Then is $a^{a}$ also transcendental number ? I've not idea about how to do it. I got motivation for asking this question from the fact that $e^...
7
votes
1answer
45 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
47 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 ...
0
votes
1answer
71 views

Monic irreducible polynomials over infinite field

If $F$ is a countable field, then proving that $F$ has algebraic closure is quite simple: there can be at most countable number of monic irreducible polynomials over $F$, let they form the set $\...
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 ...
0
votes
1answer
46 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 $F$.
1
vote
2answers
38 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
84 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 $a$ ...
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vote
2answers
60 views

Algebraic closure vs Real closure

I have proved that the surreal numbers have the properties of a real closed field. Now I should be able to explain what the importance of this real closure is. unfortunately I do not have a background ...
3
votes
2answers
360 views

Is the class of algebraic extensions distinguished? [duplicate]

In paragraph V.1 of Algebra proposition 1.7 Lang claims that the class of algebraic extensions is distinguished. I know that if $F/k$ and $E/F$ are algebraic extensions than so is the $E/k$ - that is ...
17
votes
3answers
2k views

Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem. If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
2
votes
2answers
38 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 $deg(f)=3$. ...
12
votes
1answer
248 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
1
vote
1answer
41 views

If $u, v$ have different minimal polynomials, then $F(u)$ is not isomorphic to $F(v)$?

Is the following true? Let $F$ be a field. Suppose $u,v$ have different minimal polynomials $p_u,p_v\in F[X]$, then $F(u)$ is not isomorphic to $F(v)$ as fields. I am asking this because I ...
3
votes
0answers
33 views

Do analytic properties hold in an arbitrary ordered field?

Given an ordered field, we can view it as a field formally equipped with some analytical notions which come from $\mathbb R$, like order and derivative. So I'm curious if $F$ also carries some ...
1
vote
1answer
40 views

How many polynomials are squarefree?

Of course, this depends on the field, and how we measure "how many," but it seems I cannot find an answer to this except over finite fields. My question specifically is If we have a field $F = \...
4
votes
4answers
85 views

Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

The usual answer will go like this: Since $2 \not | \ 3$ and $\mathbb{F}_{p^r} \subset \mathbb{F}_{p^s}$ if and only if $r | s$, then $\mathbb{F}_9 \not\subset \mathbb{F}_{27}$. However, I'm ...
2
votes
2answers
266 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
3
votes
1answer
117 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
3
votes
0answers
161 views

Fixed points of automorphism in the field $\mathbb{C}(x,y)$

I am trying to solve a problem, and one of the parts is the following: Let $M=\bigl(\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr)$ be a non singular $2\times 2$ matrix with integer ...
0
votes
3answers
62 views

Find a splitting field of $x^2 + 1$ over $\mathbb{Z}_3$

We know that $x^2 -3$ is irreducible in $\mathbb{Q}[x]$. We also know that $\sqrt{3}$ solves $x^2 - 3$. As a result, $x^2 - 3 = (x-\sqrt{3})(x+\sqrt{3})$. This implies that the splitting field of $x^2 ...
0
votes
2answers
47 views

Why $\mathbb{Z}_p$ can't have proper subfields?

From the notes I'm studying from I read that " $\mathbb{Z}_p=\mathbb{F}_p$ has no proper subfield." The rationale is: "assuming $\mathbb{K}$ is a subfield of a finite field $\mathbb{Z}_p= \mathbb{F}...
1
vote
1answer
62 views

Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
0
votes
1answer
108 views

Number of subrings of a Field

let $p$ be a prime number; then how many distict subrings (with unity) of cardinality $p$ does the field $F_{p^2}$ have? 1.$0$ 2.$1$ 3.$p$ 4.$p^2$ I think it will have only one subring of ...
1
vote
1answer
47 views

prove that $x^{2\cdot 3^n}+x^{3^n}+1$ is not primitive (mod 2)

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, It seems that there are no primitive polynomial of the form $$x^{2k}+x^{k}+1\quad k>1$$ and I'...
5
votes
1answer
41 views

When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
3
votes
2answers
80 views

Minimal polynomial of $\alpha = \cos\left(\frac{\pi}{48}\right)$ over $\mathbb Q$

This is a homework problem, so just a nudge in the right direction would be great. So I am required to show that $\alpha$ is a algebraic over $\mathbb Q$ and show that the degree of its minimal ...
2
votes
1answer
32 views

Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
0
votes
3answers
79 views

If $F$ is a field, what does the notation $F(x)$ mean?

If $F$ is a field, what does the notation $F(x)$ mean? I am trying to understand transcendence degree of field extension, and I am stuck in this notation. More context: I am reading this pdf, and my ...
5
votes
0answers
120 views

Points with power distances

It's possible for seven points to be at integral distances. I'm dallying with powered triangles, though, so I'm looking for point sets where all distances are powers of a fixed $x$ value. For example,...
0
votes
0answers
45 views

Finding the Galois group of $x^4+5x^2+5$

Find the Galois group of $f(x)=x^4+5x^2+5\in \mathbb{Q}[x]$. This is solved here, Exersice 3: https://math.berkeley.edu/~serganov/114/solhwg.pdf I have a question about it (I will not write all the ...
3
votes
2answers
63 views

Degree of Splitting Field to Prove Irreducibility

Let $f(x) \in F[x]$ have degree $n>0$ and let $L$ be the splitting field of $f$ over $F$. Show that if $[L:F]=n!$ then $f(x)$ is irreducible over $F$. My approach: I attempted to prove the ...