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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0answers
121 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 ...
1
vote
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
334 views

What is the meaning of $\mathbf{Q}(\sqrt{2},\sqrt{3})$

I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$?
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-...
2
votes
1answer
32 views

$\text{deg}(f)$ is not divisible by $[L:F]$

I am trying to recall an exam question so I am sorry if this question doesn't make full sense. I think some people would know what the actual wording should be after reading it. $F \subseteq L$ is ...
1
vote
3answers
60 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
2
votes
2answers
57 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: ...
1
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 $...
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 $x$. ...
1
vote
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(...
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. $rt(R)$ --> ...
3
votes
4answers
116 views

minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...
1
vote
2answers
93 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
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 ...
6
votes
2answers
69 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 ...
0
votes
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. $L=K$
1
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 ...
6
votes
2answers
149 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
votes
2answers
57 views

Is $K[a,b]=K(a,b)$ for algebraic $a, b$?

Consider a field extension $L/K$ and $a,b \in L$ algebraic over $K$. Is it true that in this setting $K[a,b]$ is already a field? I know that $K[a]=K(a)$ if and only if $a$ is algebraic over $K$.
0
votes
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 ...
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
46 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
1answer
23 views

Tower of fields - Normal [closed]

I need to construct a tower $$k \subseteq K \subseteq L,$$ such that $K/k$ is normal and $L/K$ is normal, but $L/k$ is not normal.
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
42 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 ...
1
vote
2answers
47 views

Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field.

Given Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field of $3$ elements then, $F$ is a field with $27$ elements. $F$ is separable but not a normal extension of $F_3$. The ...
4
votes
2answers
57 views

What will the underlying group of a field be isomorphic to?

Let $(F,+,.)$ be a finite field with 9 elements. Let $G=(F,+)$ and $H=(F\setminus \{0\},.)$ denote the underlying additive and multiplicative groups. Then what will $G$ and $H$ be isomorphic to? We ...
0
votes
1answer
18 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 $f(x)$ ...
0
votes
0answers
13 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(\...
14
votes
3answers
3k views

Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
4
votes
2answers
135 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 $\...
5
votes
2answers
118 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^{...
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 ...
2
votes
3answers
95 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}(...
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 $v \in E/K$ ...
1
vote
1answer
44 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{...
5
votes
3answers
41 views

Why if $E$ has characteristic $p$ then $F=\{a\in E:a^{p^n}=a\}$ is a subfield?

We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all ...
1
vote
1answer
29 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
47 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
31 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
45 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 ...
3
votes
2answers
89 views

Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
1
vote
2answers
312 views

Polynomials over infinite field [closed]

Let $K$ be an infinite field, $f$ and $g$ are two polynomials on $K$. Show that if $f\left(a\right)=g\left(a\right)\forall a\in K$, then $f=g$
1
vote
2answers
57 views

Subfields of $\mathbb{C}$ which extend to $\mathbb{C}$ via finite extensions

The field Q is a subfield of C but it is in a sense "much smaller" than C. The field R however has a finite extension of order just two to the field C. My question is: are there other subfields of C ...