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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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
135 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
89 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
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2answers
274 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 ...
1
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3answers
68 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
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1answer
123 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 ...
0
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2answers
49 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
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1answer
69 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 ...
3
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1answer
58 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'...
3
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2answers
83 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
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3answers
81 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
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0answers
49 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
130 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
34 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 ...
1
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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 $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
5
votes
1answer
42 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
60 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
26 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
36 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: ...
1
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2answers
68 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
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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
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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 $x$. ...
0
votes
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. $rt(R)$ --> ...
1
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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 ...
1
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2answers
95 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
vote
1answer
49 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
73 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. ...
0
votes
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 ...
1
vote
3answers
174 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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votes
2answers
151 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 ...
0
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2answers
64 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
89 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 ...
0
votes
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 ...
1
vote
1answer
48 views

Prove there is no such nth root of unity $\zeta$ such that $\mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\zeta) \quad$ [duplicate]

I'm trying to do the above problem. My approach is to use the fact that $\mathbb{Q(\zeta)}$ is the fixed subfield of the normal subgroup $A_3$ of $S_3$ and then since $A_3$ has no subgroup of the form ...
0
votes
0answers
43 views

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$. Suppose that $f$ is a irreducible polynomial of ...
0
votes
0answers
14 views

Separable extensions are distinguished

I'm studying Steve Roman's book "Field Theory" and I found this proof about separable extensions being distinguished but I don't understand his proof. More exactly, why does he conclude from $F<F(\...
0
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1answer
20 views

Knowing the Galois group of the splitting field of a polynomial $f$, how can I show that $f$ is irreducible in the ground field?

So I'm given $f(x) = \sum_{k=0}^{8}\frac{x^k}{k!} \in \mathbb{Q}[x]$. Denote its splitting field by $E$, then I'm also given that ${\rm Gal}(E/\mathbb{Q}) \cong A_8$. The task is to prove that $f(x)$ ...
3
votes
1answer
42 views

What does $K(A)$ mean in field theory?

So in my notes it says that if $K\subset L$ is a field extension and $A \subset L$ is a subset then $K(A)$ is a subfield of $L$ containing both $K$ and $A$. It is in fact the smallest such subfield. I ...
0
votes
0answers
20 views

Given a tower of extensions how to show that the degree of an extension is even?

Suppose $\Bbb{Q} \subseteq F \subseteq \Bbb{C}$ is a tower of extensions and suppose that $i \in F$. If the extension $\Bbb{Q} \subseteq F$ is finite, show that $[F : \Bbb{Q}]$ is even. What ...
4
votes
2answers
137 views

Is $\sqrt{7} \in \mathbb{Q}(\sqrt{3+\sqrt{2}})\;$?

Let $u = \sqrt{3+\sqrt{2}}\;$. Is $\sqrt{7} \in \mathbb{Q}\left(u\right)$? Equivalently, is $\mathbb{Q}(u)$ a splitting field of $u$ over $\mathbb{Q}\,$? The original question is whether or not $\...
2
votes
3answers
96 views

Prove $2^{1/2}+3^{1/3}$ is irrational using Galois theory.

So, I want to prove that $2^{1/2}+3^{1/3}$ is irrational, and I need to prove it using Galois theory. To start, let's forget about the sum and deal with the individual numbers and $F_1 = \mathbb{Q}(...
1
vote
1answer
45 views

How can I show that the Galois group of $x^p -1$ is abelian?

So $p$ is prime, and we have $f = x^p -1 \in \mathbb{Q}[x]$ with splitting field $E$. I need to show that ${\rm Gal}(E/\mathbb{Q})$ is abelian and of order $p-1$. The splitting field $E$ is $\mathbb{...
0
votes
0answers
20 views

Let $F=K(u)$ where $u$ is transcendental over $K$, prove that it is algebraic over $E$, where $K \subset E \subseteq F$

Let $F=K(u)$ where $u$ is transcendental over $K$. Prove that it is algebraic over $E$, where $K \subset E \subseteq F$. The method I tried for the above question was as follows: Choose $v \in E/K$ ...
5
votes
2answers
121 views

Neat method to show that $\mathbb{Q}(2^{1/3}) \ne \mathbb{Q}(3^{1/3}) $?

I am wondering how to show looking obvious $\mathbb{Q}(2^{\frac{1}{3}}) \ne \mathbb{Q}(3^{\frac{1}{3}}) $? This question has appeared to compute the order of $\text{Gal}(\mathbb{Q}(2^{\frac{1}{3}},3^{...
1
vote
1answer
31 views

About separable extensions (one more time)

Well I'm stuck trying to prove the following about separable extensions. If $L/E$ is a extension (not necessarily finite) such that $L/F$ and $F/E$ are both separables, then $L/E$ is also separable. ...
2
votes
1answer
48 views

Galois Group Isomorphic to $S_3$.

Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree 3. Suppose $f$ has one real root, we want to show that $$\text{Gal}(L/\mathbb{Q}) \cong S_3,$$ where $L$ is the splitting field of $f$. ...
1
vote
1answer
34 views

Field Extension for which Galois correspondence fails [closed]

Find a non-Galois field extension such that the Galois correspondence fails. Can't seem to come up with a nice answer to this.
0
votes
2answers
47 views

I need help understanding a proof (Kronecker's theorem)

Kronecker's theorem says that if $F$ is a field and $f(x)$ is a non-constant polynomial in $F[x]$, then there exists an extension field $E$ of $F$ in which $f(x)$ has a root. Here's the proof ...
6
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0answers
85 views

Abelian groups whose finite subgroups are cyclic

If $(F,+,\times)$ is any field, then the abelian group $(F-\{0\},\times)$ has property that every finite subgroup of it is cyclic. Question: If $G$ is an abelian group such that every finite subgroup ...