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 ...

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-1
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0answers
24 views

$g(x) = f(x) \gcd(f,f')^{-1}$

In a field of characteristic 0, $F$, if $f(x) \in F[x]$ is monic with positive degree and $d(x)$ = gcd($f(x),f'(x)$) (i.e. the derivative), then $g(x) = f(x)d(x)^{-1}$ has the same roots as $f(x)$. ...
3
votes
2answers
76 views

Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...
0
votes
1answer
19 views

The elements in the composite field FK

Where F,K are two fields. What does the element in FK look like? All the elements are generated by the elements of F and K?(combination of the elements of F and K) I think FK is pretty close to the ...
1
vote
1answer
25 views

Real Algebraic Numbers is Real Closed

I want to show that if $R$ is a real closed field, then its subfield of elements which are algebraic over $\mathbb{Q}$ is real closed. Let $R$ be real closed. Let $R_0$ be the subfield of elements ...
1
vote
0answers
24 views

prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$

$\mathbb{Q}<\mathbb{F}<\mathbb{C}$ - field extensions, such that $[\mathbb{F}:\mathbb{Q}]=m \in \mathbb{N}$ p is a prime number, $\alpha=p^{\frac{1}{n}}$ gcd(m,n)=1 prove n divides ...
7
votes
0answers
70 views

Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by ...
3
votes
1answer
67 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring ...
2
votes
1answer
22 views

Automorphisms of a field extension (proof verification)

I am asked to compute the automorphisms of the field extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$. I know that $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=4$ since $$ ...
1
vote
1answer
29 views

Cyclotomic Fields - Showing that the fixed field of $G(\mathbb Q(\xi)/\mathbb Q)$ is $\mathbb Q$.

If $p$ is a prime and $\xi$ is a primitive $p$th root of unity, I know that $G(\mathbb Q(\xi)/\mathbb Q) = \{\psi_{\xi,\xi^k}\}_{1\leq k<p}$, where for each $k$, $\psi_{\xi,\xi^k}(\xi) = \xi^k$. I ...
1
vote
0answers
27 views

Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
0
votes
1answer
14 views

Definition of normal extension.

Let $E/F$ be an algebraic extension. Then, $E/F$ is normal iff $E$ is a splitting field of a family of polynomials in $F[X]$. So does this mean that if $E$ is a splliting field of a given ...
-1
votes
3answers
37 views

How are all the roots of unity of cyclotomic extension are of this form? [closed]

Suppose $x \in Q(\zeta_n) $ which satisfy $x^t =1, t \in \mathbb{N}$. Then show that $x$ is of the form $\zeta_n^k$ for some $k$ where $1 \leq k \leq n-1$ ?
0
votes
1answer
30 views

an “explicit” extension field that contains a root of an irreducible polynomial

There is a famous theorem saying that Let $\Bbb F$ be a field and $f(x)$ an irreducible polynomial in $\Bbb F[X]$. Then there exists a field extension $\Bbb L$ of $\Bbb F$ such that $f(x)$ has a ...
2
votes
1answer
48 views

Determine whether the extension is Galois [duplicate]

I am trying to prove that $K=\mathbb{Q}(2^{1/3}, i\sin{2\pi/3})$ is Galois extension over $\mathbb{Q}$. It is easy to see that $K=\mathbb{Q}(2^{1/3},i\sqrt{3})$. I know it is Galois since $K$ is a ...
0
votes
0answers
12 views

Question on degree of field extension

Hello all I have the following question in field theory HW assignment and would really appreciate any help. We are given a field F and two extensions of it such that the following holds: ...
0
votes
1answer
25 views

About the additive group and the multiplicative group of a field

Let $F$ be a field. When happens that the additive group of $F$ is isomorphic to the multiplicative group? It is easily to work out that $F$ must have characteristic $0$, but then what?
1
vote
2answers
32 views

Why does $F[\alpha]=F(\alpha)$ imply $\alpha$ is algebraic?

Let $E/F$ be a field extension. Let $\alpha\in E$ such that $F(\alpha)=F[\alpha]$. Then, how do I prove that $\alpha$ is algebraic over $F$?
2
votes
1answer
53 views

Is every isomorphism of an algebraically closed field onto one of its subfields an automorphism?

I've just been reading about the Isomorphism Extension Theorem, and I think I can make the following argument: Let $F$ be an algebraically closed field, and let $\sigma$ be an isomorphism of $F$ onto ...
2
votes
3answers
77 views

Let $\alpha\in R$. Prove that $\mathbb Q(\alpha)\cong\mathbb Q(x)$ iff $\alpha$ is transcendental.

Let $\alpha\in\mathbb R$. Prove that $\mathbb Q(\alpha)\cong\mathbb Q(x)$ if and only if $\alpha$ is transcendental So is this saying that we need everything (all expressions involving x) to ...
1
vote
2answers
33 views

A question on on field extensions and minimal polynomial

Hello I am a novice in fields and was asked this question in an assignment: I need to find the minimal polynomial of the expression $\sqrt[3]{7-\sqrt{2}}$ over the rationals Q Here is where I am ...
4
votes
0answers
47 views

What is the automorphism group of the field of all constructible numbers?

Let $\Omega\subseteq \mathbb{C}$ be the field of all constructible numbers (i.e. $\Omega$ is the smallest subfield of $\mathbb{C}$ which is closed under taking square roots). What is known about the ...
0
votes
0answers
32 views

If L/K is algebraic field extension, is L(B)/K(B) also algebraic?

Seems a naive question, but I get stuck there.. Say $M/L/K$ is a tower of fields, $L/K$ is algebraic, $B\subset M$. ($B$ is not necessarily algebraic independent over $K$ or $L$.) Is $L(B)/K(B)$ ...
1
vote
2answers
29 views

Minimal polynomial problem

Show that $\mathbb Q(\sqrt 2 + i)=\mathbb Q(\sqrt 2, i)$ and find minimal polynomial. My question: Assume that they are equal, then the minimal polynomial of both sides must be the same. To prove and ...
0
votes
0answers
42 views

A simple question in field theory from a beginner

I am a beginner in field theory and have the following assignment question which I need help on: I am asked to prove that the following equality holds: $$\mathbb Q(\sqrt{5} , \sqrt{7}) = \mathbb ...
1
vote
0answers
52 views

Cyclotomic polynomial, after adjoining a radical

Suppose that $p>2$ is prime and that $a$ is a rational number for which $\sqrt[p]{a}$ is in $\mathbb C\backslash\mathbb Q$. The cyclotomic polynomial $\Phi_p$ is well-known to be irreducible over ...
0
votes
0answers
20 views

If $S=\sum (\frac{n}{p})\zeta^n$ then how to prove that $S^2=(\frac{-1}{p})p$? [duplicate]

Here $\zeta$ is a primitive $n$-th root of unity and ($\frac{n}{p}$) denotes the Legendre symbol. Can someone please give a proof of this fact? I tried writing $S^2$ as the product of two sums $S=\sum ...
2
votes
2answers
88 views

Suppose that $a$ and $b$ belong to a field of order $8$ and that $a^2 + ab + b^2 =0$ then $a=0$ and $b=0$ . [duplicate]

Suppose that $a$ and $b$ belong to a field of order $8$ and $a^2 + ab + b^2 =0$. Then $a=0$ and $b=0$. Do the same when the field has order $2^n$ with $n$ odd? If one of the term is zero, i.e. ...
4
votes
1answer
47 views

Prove the extension to be a Galois Extension

Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension. Trial: Since this $\mathbb C$ is a field of charactersitic $0$,it would be ...
1
vote
1answer
26 views

In this proof, why is $\gamma[N]$ a proper subset of R/M?

I have highlighted what I do not get in red: I need that $\gamma[N]$ is a proper subset of $R/M$. Why is it that we have this? I get that $\{0+M\}$ is a proper subset of $\gamma[N]$, since the ...
1
vote
1answer
26 views

Simple algebra over algebraically closed field

In Jacobson's Lie Algebras, page 303, it seems he uses the following result: If $\mathfrak L$ is a simple finite-dimensional Lie algebra over a field $\Omega$ which is the algebraic closure of a ...
1
vote
0answers
23 views

Does $[F(\zeta_n) : F ]$ divide $\phi(n)$?

I know that if $F=\mathbb Q$, the degree actually equals $\phi(n)$. Also, if the extension $F(\zeta_n)/F$ is Galois, then I can invoke my knowledge of the existence of an injective map from ...
2
votes
0answers
28 views

Show that K is a splitting field for some degree 4 polynomial f(x) in k[x].

Suppose $K/k$ is a finite Galois extension such that the Galois group of $K/k$ is isomorphic to $\mathcal S_4$. How can we show that $K$ is a splitting field for some degree $4$ polynomial $f(x)$ in ...
1
vote
0answers
14 views

Show that invertible elements of the algebraic closure of $F_p$ is not cyclic

I want to show that invertible elements of the algebraic closure of $F_p$ is not cyclic, where p is a prime. I know that the algebraic closure of $F_p$ is countably infinite, since it is equal to the ...
0
votes
1answer
50 views

What are some examples of unconventional fields?

We started talking about fields in my foundations of mathematics class, and since the symbols we are using are + and •, I keep catching myself giving them properties of multiplication and addition. ...
0
votes
0answers
12 views

How do I prove that there is a unique such finite subfield?

Let $E$ be an algebraically closed field. Let $F,L$ be finite subfields of $E$ such that $F\cong L$. Then, how do I prove that $F=L$??
1
vote
1answer
31 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
0
votes
0answers
11 views

Name of $\bar{F_E}$?

Let $E/F$ be a field extension. Define $\bar{F_E}:=\{\alpha\in E : \alpha \text{ is algebraic over } F\}$. What is the standard name of this? Fraleigh calls it 'the algebraic closure of $F$ in ...
1
vote
2answers
34 views

Factorization of polynomial in a complete field

Let $k$ be a finite extension of $\mathbb{Q}$ and $|\cdot|$ an absolute value on it (either Archimedean or not). Let $L$ be the completion of $k$ with respect to this value, and take any irreducible ...
4
votes
3answers
488 views

Why do they need commutativity in the proof?

In this proof they prove it for fields, but they say that the proof holds for a commutative ring with unity. But why do we need it to be commutative, and why do we need a unity? Why does it not hold ...
0
votes
1answer
38 views

Question about $[K:F]=[K:E][E:F]$ for fields $E,F,K$

This is probably a dumb question. We have a theorem that states If $E$ is a finite extension field of a field $F$, and $K$ is a finite extension field of $E$, then $K$ is a finite extension of ...
0
votes
0answers
24 views

Let F be a field and let a,b ∈ F with a not equal to b. Show that the polynomials f ( x ) = x + a and g ( x ) = x + b are relatively prime

Let F be a field and let $a,b ∈ F$ with $a$ is not equal to $b$ . Show that the polynomials $f ( x ) = x + a $ and $g ( x ) = x + b $ are relatively prime. Only things I know is : A commutative ring ...
0
votes
1answer
72 views

Complex number field: ''essentially'' unique?

I solved the following exercise but have trouble making sense of the result: If $\widetilde{\mathbb C}$ is another field of complex numbers and $\varphi : \mathbb C \to \widetilde{\mathbb C}$ is a ...
0
votes
2answers
26 views

If $x^3 + x + 1$ is reducible over $F$, then why does it follow that $E\subseteq F$?

I'm trying to understand a solution to the following problem: Show that $K = \mathbb Z_2[y,z]/\langle y^4 + y^3 + 1,z^3 + z + 1\rangle$ is a field. A solution is as follows (the part I don't ...
0
votes
0answers
51 views

A way to sum supernatural numbers involving Zeta function's analytic continuation

I have this idea on how to sum supernatural numbers assigning them a finite value in a way similar to how we assume that the sum of every natural numbers from 1 to infinity equals $-\frac 1 {12}$. ...
22
votes
2answers
809 views

Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac ...
3
votes
1answer
39 views

A Field has characteristic 0 if and only if it contains Q

I would like some help in proving the following statement: A field $K$ has characteristic 0 if and only if $\mathbb{Q}$ is a subfield of $K$. So, the ...
5
votes
1answer
34 views

Inverting the elements of a basis of a finite field extension

Let $K/k$ be a finite field extension of degree $d$. Suppose that $\{a_1, \dots, a_d\}$ is a basis of $K$ as a $k$-vector space. Is it true that $\{a_1^{-1}, \dots, a_d^{-1}\}$ is a basis of $K$ as a ...
2
votes
1answer
45 views

Roots of a polynomial and finite additive subgroup

Suppose $F$ is a field with characteristic $p$ and $f(x)\in F[x]$ Then$f(x)=x^{p^m}+a_1x^{p^{m-1}}+\cdots +a_mx \iff$ its roots form a finite subgroup of the additive group of the splitting field. ...
2
votes
1answer
63 views

Transcendence Degree of a field extension over $\mathbb C$

Consider the $2 \times n$ matrix $\begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2n} ...
0
votes
1answer
25 views

How is the degree of the minimal polynomial related to the degree of a field extension?

I was reading through some field theory, and was wondering whether the minimal polynomial of a general element in a field extension L/K has degree less than or equal to the degree of the field ...