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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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{...
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0answers
20 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, ...
6
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1answer
42 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
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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 ...
4
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0answers
26 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
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
82 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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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 $F$.
2
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2answers
37 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 $deg(f)=3$. ...
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2answers
56 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 ...
1
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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
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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 ...
0
votes
1answer
70 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 $\...
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 = \...
8
votes
1answer
127 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^...
4
votes
4answers
82 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
265 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 ...
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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
1answer
95 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 ...
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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}...
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0answers
17 views

Mention a concrete example of a field (like $\mathbb{R}$) [closed]

This example must not be complete, I'm trying to understand the contrast between those two kind of fields, this is the problem, I can't even find an example. Thank you!.
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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 ...
1
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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'...
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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 ...
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 ...
0
votes
0answers
44 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 ...
5
votes
0answers
119 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,...
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 ...
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2answers
86 views

What does $\mathbb{Q}(\sqrt{2},\sqrt{3})$ mean? [duplicate]

What set is $\mathbb{Q}(\sqrt{2},\sqrt{3})$? Is it the set $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
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1answer
40 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 ...
2
votes
2answers
55 views

Number of solutions a polynomial can have as a function of the field?

Is there any limitation (upper bound) for number of solutions of polynomial equations? Having a background in engineering, my knowledge of higher algebra is rather limited, but I do know of ...
0
votes
1answer
25 views

Number of Subfields sandwiched between two fields

Let $\omega$ be a complex cube root of unity such that $\omega \neq 1$. Suppose L is the field $\mathbb Q(2^{1/3},\omega)$ generated by them over the field of rationals. Then, the number of subfields ...
0
votes
1answer
35 views

A reduction to the finite degree case

I am stuck trying to understand a proof in Asymptotic Differential Algebra and Model Theory of Transseries by L. van den Dries, J. van der Hoeven and M. Aschenbrenner. The result is the following: ...
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vote
2answers
67 views

If $\mathbb{F}$ is a finite field such that for every $a \in \mathbb{F}$ the equation $x^2=a$ has a solution in $\mathbb{F}$

Let $\mathbb{F}$ be a finite field such that for every $a \in \mathbb{F}$ the equation $x^2=a$ has a solution in $\mathbb{F}$. Then $\mathbb{F} \simeq \mathbb{F}_{2^n}$. I have tried this for the ...
1
vote
1answer
28 views

Ring Homomorphisms of Fields with $f(1)=1$ are Injective?

True or False: If $F_1, F_2$ are fields and $f:F_1\to F_2$ is a ring homomorphism such that $f(1)=1$, then $f$ is injective. I am not sure if this is true. Here's an attempt at a counter. Consider $...
2
votes
1answer
98 views

Luröth's Theorem

I've been struggling trying to understand the Jacobson's Basic Algebra vol. II proof of the Luröth's theorem. Let $K$ be a field, $K(X)$ the field of rational fonctions and take $L$ to be a sub-...
0
votes
0answers
23 views

Field extension over the rationals does not have a square root of -$\alpha^2$

Let $f=x^4-2\in\mathbb{Q}[x]$ and consider the field $K=\mathbb{Q}[x]/(f)$. I want to show that There exists no element $u\in K$ such that $u^2=-\alpha^2$, where $\alpha$ is the coset of $x$. ...
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2answers
30 views

Discrete Mathematics (Closure Problems)

$R = \{(x, x+1)|x \in \mathbb{Z}\}$ $\mathbb{Z}$ is the integers and could be negative or positive. Create the closure of the the following: a. $t(R)$ --> transitive closure of R b. $rt(R)$ --> ...
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0answers
48 views

Describe the separable closure of $Z_3(u,v)$ in $Z_3(y,z)$

I'm struggling with the separable closure problem and I don't understand some points. Please explain why it is.. WTS : Describe the separable closure of $Z_3(u,v)$ in $Z_3(y,z)$ Let y,z be ...
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2answers
90 views

Show that $\mathbb{Z}[i]/n\mathbb{Z}[i] $ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$.

I have to show the following statement: $\mathbb{Z}[i]/n\mathbb{Z}[i]$ is a field if and only if $n$ is a prime number and $n\neq a^2+b^2, a,b\in\mathbb{Z}$. Let $\mathbb{Z}[i]/n\mathbb{Z}[i]$ ...
1
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1answer
47 views

Intermediate fields of a finite field extension that is not separable

Let $\mathbb{F}_p$ be the finite field with $p$ elements, where $p$ is a prime number. Let $x$ and $y$ be transcendental and algebraically independent over $\mathbb{F}_p$. The extension $\mathbb{F}_p(...
6
votes
2answers
68 views

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
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0answers
13 views

Each exponent of each term of an irreducible polynomial is divisible by p

I'm studying the field theory,in particular, the separable extension. My question is the followings. WTS : an irreducible polynomial q(x) over a field F of characteristic p≠0 is not separable iff ...
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vote
3answers
170 views

On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
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2answers
147 views

Isomorphic fields of finite degree have same dimension over base field

Let $K/F$ be a field extension and $L_1,L_2$ subfields of $K$ such that $L_1$ and $L_2$ have finite degree over $F$. Does $L_1 \cong L_2$ imply $[L_1 : F ]=[L_2 : F]$? Obviously, if the isomorphism ...
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2answers
62 views

Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$, let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. [closed]

Let $\omega= \cos \frac{2\pi}{10}+i\sin \frac{2\pi}{10}$. Let $K=\mathbb{Q}(\omega^2)$ and $L=\mathbb{Q}(\omega)$. Then A. $[L,\mathbb{Q}]=10$ B. $ [L,K]=2$ C. $[K,\mathbb{Q}]=4$ D. $L=K$
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2answers
86 views

Proving $\mathbb R[x]/\langle 1+x^2\rangle$ $\cong$ $\mathbb C$ without using 1st isomorphism theorem

I've seen many the proofs of this by making use of First isomorphism theorem, by considering the map,$$\phi:\mathbb R[x]\rightarrow\mathbb C$$ defined by $\phi(a+bx)=a+bi$. My questions are ...
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2answers
31 views

If a Galois group has $n$ subgroups of some order $k$, will there always be $n$ intermediate field extensions of order $k$?

I realised today that I don't really understand the entirety of the fundamental theorem of Galois theory. It might be that the way it's phrased in my book confuses me, or it might be the subject ...