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Questions tagged [field-theory]

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. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

5
votes
0answers
24 views

Are logarithms of prime numbers quadratically independent over $\mathbb Q$?

It is well-known that the logarithms of prime numbers are linearly independent over $\mathbb Q$. It is also known that the question whether the logarithms are algebraically independent over $\mathbb Q$...
2
votes
1answer
29 views

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals.

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals. My attemt: Any general equation of degree $\geq 5$ is not solvable by radicals. ...
1
vote
1answer
36 views

Kernel of a polynomial ring homomorphism

Let $\mathbb{F}$ be a field, and define a homomorphism $\phi:\mathbb{F}[x,y]\rightarrow \mathbb{F}[z]$ by: $$f(x,y)\mapsto f(z^a,z^b)$$ where $a\neq b\in \mathbb{N}$. My question is: For what $\...
1
vote
0answers
27 views

Degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ and splitting field

I have two questions: Determine the degree of the extension degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ over $\Bbb Q$ and the degree of the splitting field of the minimal polynomial of $\sqrt{2 + \...
0
votes
3answers
30 views

Is there a cubic irreducible polynomial over $\Bbb Q$ with all of its three roots are irrational?

Prove or Disprove: For every irreducible cubic polynomial $f(x)\in \Bbb Q[x]$, there exist a subfield $F$ of $\Bbb C$ such that $F \nsubseteq \Bbb R$ and $F \simeq \Bbb Q[x]/\langle f(x) \rangle $ ...
20
votes
2answers
924 views

What fields between the rationals and the reals allow a good notion of 2D distance?

Consider a field $K$, let's say $K \subseteq \mathbb R$. We can consider the 'plane' $K \times K$. I am wondering in which cases the distance function $d: K \times K \to \mathbb R$, defined as is ...
1
vote
2answers
34 views

Does an uncountable algebraically closed field of characteristic zero contain an uncountable subfield which can be embedded into $\mathbb{C}$?

This is probably very simple. Let $k$ be an uncountable algebraically closed field of characteristic zero. Does there exist an uncountable algebraically closed subfield $k_0\subset k$ and an ...
0
votes
1answer
19 views

Prove that $S = \{A \in GL_n(K) | AJA^t = J\}$ is a subgroup of $GL_n(K)$

Given a field $K$, $n \in \mathbb{N}$ and $J \in K^{n \times n}$, show that: $$ S = \{A \in GL_n(K) | AJA^t = J\} $$ is a subgroup of $GL_n(K)$. To show that S is a subgroup, we must show: It must ...
2
votes
1answer
67 views

A field with 729 elements [duplicate]

I am looking for an explicit description of the additive, multiplicative group structure and automorphism group of the field with 729 elements. Sorry for not being clear. I have no clue what does "...
0
votes
0answers
45 views

Is it true that a finite field extension with degree $>1$ cannot be isomorphic to its base field?

Is it true that a finite field extension with degree $>1$ cannot be isomorphic to its base field? Suppose two fields are isomorphic via the isomorphism $f: E \to F$. Then it is true that using $f$ ...
2
votes
0answers
23 views

$[\mathbb{F}_{p}(\alpha,\beta)$:$ \mathbb{F}_{p}]=\text{lcm}(m,n) $ and $[\mathbb{F}_{p}(\alpha)∩\mathbb{F}_{p}(\beta)$:$ \mathbb{F}_{p}]=\gcd(m,n)$

My grammar can be awkward because my native language is not English. Please excuse me. [Problem] There are $ \alpha, \beta\in K$ that is a field extension of $ \mathbb{F}_{p}$ such that $[\...
0
votes
1answer
22 views

Normal Closure of an Algebraic Extension

$\bf{Q.}$ Let $K/F$ be an algebraic extension. Show that there is an algebraic extension $L/K$ such that $L/F$ is normal and if $M$ is another normal extension of $F$ such that $F\subseteq K\subseteq ...
1
vote
3answers
54 views

$K=\Bbb Q(\sqrt3,\sqrt[3]{2}) $, Compute $[K:\Bbb Q]$.

I've tried looking at similar examples for but something has been close enough. How do I compute $[\mathbb Q(\sqrt 3,\sqrt[3]2):\mathbb Q]$. I think you say that it's equal to $[\mathbb Q(\sqrt3 + \...
0
votes
0answers
39 views

Intermediate Galois subfields of $ \mathbb{Q}(\sqrt[n]{\alpha}) $

Let $ \alpha > 0 $ in $ \mathbb{Q} $ and let $ K = \mathbb{Q}(\sqrt[n]{\alpha}) $ of degree $ n $ over $ \mathbb{Q} $. Determine all nontrivial subfields of $ K $ that are Galois over $ \mathbb{Q} $...
0
votes
0answers
39 views

What is the use of extension of the real and complex field?

What is the use of the extension of the real and complex fields to extended real and complex numbers (including $\infty$)? If the extended real and complex sets are no longer fields, then what is the ...
0
votes
2answers
34 views

Is $\mathbb{Q}(i,\sqrt[3]{2})=\mathbb{Q}(i,\sqrt[3]{4})$?

Is $\mathbb{Q}(i,\sqrt[3]{2})=\mathbb{Q}(i,\sqrt[3]{4})$? $\mathbb{Q}(i,\sqrt[3]{2})\supseteq\mathbb{Q}(i,\sqrt[3]{4})$ holds because $\sqrt[3]{4}=(\sqrt[3]{2})^2$, but I don't know how to show the ...
0
votes
0answers
24 views

Field is perfect iff every element has pth root.

I am trying to understand the proof for one direction of the following theorem: I am confused about the part in red. Why does this work? Has it something to do with the Frobenius endomorphism?
1
vote
1answer
36 views

Infinite perfect field of characteristic p [duplicate]

Given a perfect field of prime characteristic $p$, is it necessarily finite? I believe there must be some counterexample. However, the only infinite field of characteristic $p$ that I know of is $\...
4
votes
2answers
25 views

Let $K$ be an extension of a field $F$ such that $[K:F] = 13$.Suppose $a$ $∈$ $K-F$.What is the value of $[F(a):F] $?

Since 13 is prime and we can write $[K:F]$ $=$ $[K:F(a)]$ $[F(a):F]$ , i think answer should be 1 or 13.
6
votes
0answers
42 views

First-order definability sums of squares

Let $K$ be a field. I am interested in when there can exist a first-order definition of the set $$ \Sigma K^2 := \lbrace \sum_{i=1}^n x_i^2 \mid n \in \mathbb{N}, x_1, \ldots, x_n \in K \rbrace $$ in $...
0
votes
1answer
16 views

Example of an infinite algebraic field extension which is not simple

I want to an example of an infinite algebraic field extension that is not simple. I was thinking of the algebraic numbers $\mathbb{A}$ over $\mathbb{Q}$, or all the square roots of primes adjoined to $...
1
vote
1answer
15 views

Suppose $\zeta$ is a root of unity and $f\in \mathbb{Q}[X]$. Show that $f(\zeta)\neq2^{1/4}$.

Question: Suppose $\zeta$ is a root of unity and $f\in \mathbb{Q}[X]$. Show that $f(\zeta)\neq2^{1/4}$. This is from the UCLA fall '16 algebra qual. So far I haven't gotten far besides my initial ...
0
votes
0answers
15 views

(Intermediate) normal extension is stable

My questions is about this famous result: Let $L\supset N \supset F$ be a tower of field extensions where $N$ is normal. For any $\sigma \in Gal(L/F), \sigma(N)=N$. The following is what I have ...
1
vote
1answer
33 views

Let $L/K$ be a field extension and let $a, b \in L$ be algebraic elements over $K$ having the same minimal polynomial. Show that $K(a) \simeq K(b)$.

The above question was given to me as an assignment but I'm a bit stuck and I think the way I've been going about it is a dead end, or if not a dead end, I can't figure out a way to finish it off So ...
0
votes
1answer
20 views

Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?

Exercise sounds: Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?I have the solution (on picture). Is it correct? Why do we prove in this way, why we must show that square, ...
1
vote
0answers
33 views

How to construct a given group as multiplicative group of field

Question: Assume an abelian group $G$ is given. Then, is there a field $F$ such that $F^{\times} \cong G$ ? What is the necessary and sufficient condition about G? I already know : If $G$ is ...
0
votes
0answers
28 views

Prove if two normal extensions over a field are isomorphic

Suppose $K_1,K_2$ are normal extensions over a field $F$ . Suppose the set of minimal polynomials of elements in $K_1$ over $F$ = the set of minimal polynomials of elements in $K_2$ over $F$ . Then ...
0
votes
0answers
21 views

Number of embeddings of a (nonsimple) extension of a field to another field

Let $L\supset F$ be a finite field extension of degree $n$ and $K \supset F$ be any extension. I wonder how to prove that the number of embeddings from $L$ to $K$ that restricts to identity on $F$ is ...
0
votes
1answer
44 views

Show that $n\in\mathbb{N}$ is square-free

Show that $n\in\mathbb{N}$ is square-free if and only if there is a subset different from the null set, $L\subseteq\mathbb{Z}_n$, with the property that summation and multiplication from $\mathbb Z_n$ ...
-4
votes
0answers
38 views

Field and abelian group [duplicate]

Are the following two statements equivalent to each other ? 1) $(R,+,*)$ is a field 2) $(R,+)$ is an abelian group and $(R\setminus\{0_R\},*)$ is an abelian group If not give an example.
1
vote
1answer
22 views

If $u\in F$ is Transcendental over $K$, $F$ an extension field of $K$, Show every element in $K(u)$ not in $K$ is transcendental over $K$.

Doing some problems out of Beachy’s Algebra text, I came across that problem, and I’m at a loss how to show it without a bit of hand waving. Do I make some statement about spaces, and prove by ...
0
votes
0answers
25 views

Compute the size of the orbit under a finite group action

When $n = 3$, consider the three-dimensional tensors $X, X' \in \mathbb{F}_2^{2^n}$ given by $X = A \otimes e_1 + B \otimes e_2$ and $X' = B \otimes e_1 + (A+B) \otimes e_2$ where $A = \begin{...
2
votes
1answer
40 views

Group automorphism of multiplicative group of real number field

Let $\mathbb{R}$ be the real number field and $\mathbb{R}^{\times}$ be the multiplicative group of it. $\mathrm{Aut}(\mathbb{R}^{\times})$ denotes the group automorphism of $\mathbb{R}^{\times}$. [...
4
votes
1answer
78 views

Extension of Splitting Fields over An Arbitrary Field

Let $F$ be a field in which $0 \neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact?...
1
vote
2answers
29 views

Let $k$ be a field and $x,y \in K[x,y]$, then $x$ and $y$ are relatively prime.

Let $k$ be a field and $x,y \in K[x,y]$, then $x$ and $y$ are relatively prime. I already proved that 1 is cannot be a linear combination of $x$ and $y$. I have already tried this directly. I ...
2
votes
1answer
46 views

Marcus Number fields exercise 17 chapter 4

I know this question has already been posted, but I don't manage to understand the comments. Exercise $17$ $(e)$ on Marcus' Number Fields, Chapter $4$ My problem is exactly the same as the one ...
0
votes
0answers
20 views

If $F(x) \cong \text{Frac}(R[x])$ where $F=\text{Frac}(R)$ so I want prove that $F(x_{1},…,x_{n}) \cong \text{Frac}(R[x_{1},…,x_{n}]) $ .

I already proved that $$F(x) \cong \text{Frac}(R[x])$$ where $F=\text{Frac}(R)$ so I want prove that $$F(x_{1},...,x_{n}) \cong \text{Frac}(R[x_{1},....,x_{n}])$$ Before trying this by induction I ...
0
votes
4answers
80 views

Let $F$ be a field. (a) If $1 + 1 = 0$, show that $a + a = 0$ for all $a \in F$. (b) If $a + a = 0$ for some $a \neq 0$, show that $1 + 1 = 0$

Let $F$ be a field. (a) If $1 + 1 = 0$, show that $a + a = 0$ for all $a \in F$. (b) If $a + a = 0$ for some $a \neq 0$, show that $1 + 1 = 0$ I have found proof's for $1+1=0$ but I am not ...
3
votes
1answer
32 views

Let $M$ be a subfield of Complex field such that $M/\Bbb Q$ is a finite Galois extension

Let $M$ be a subfield of Complex field such that $M/\Bbb Q$ is a finite Galois extension. Show that if $[M:\Bbb Q]$ is an odd number, then $M$ is a subfield of Real field. My current thought is since ...
1
vote
1answer
55 views

Frac$(R)=F$, quotient field of the integral domain $R$, then Frac$(R[x]) \cong F(x)$ and also Frac$(R[x]) \cong F(x_{1},x_{2},…,x_{n})$.

Let $R$ be an integral domain, and consider $\operatorname{Frac}(R)=F$ the quotient field of $D$. Then $\operatorname{Frac}(R[x]) \cong F(x)$ and also $\operatorname{Frac}(R[x]) \cong F(x_{1},x_{2},......
0
votes
1answer
41 views

Fields and Groups equivalent [closed]

Are the following two statements equivalent to each other ? 1) $(R,+,*)$ is a field 2) $(R,+)$ is an abelian group and $(R\setminus\{0_R\},*)$ is an abelian group If not give an example.
3
votes
3answers
123 views

Is it possible to have an Abelian group under two different binary operations but the binary operations are not distributive?

I am trying to show that if $(R, +)$ is an Abelian group and $(R - \{0_R\}, \cdot)$ is an Abelian group, then $(R, +, \cdot)$ is not necessarily a field. Note that $0_R$ is the identity element of $(R,...
1
vote
1answer
31 views

Show that $\left< \mathbb{Q}\left[\sqrt 2\right], < \right> \simeq \left<\mathbb{Q}, < \right>$

Show that $\left< \mathbb{Q}\left[\sqrt 2\right], < \right> \simeq \left<\mathbb{Q}, < \right>$, where $\mathbb Q$[$\sqrt 2$] is the smallest subfield of $\mathbb R$ containing $\...
-2
votes
1answer
33 views

A finite ring which contains a field [closed]

Prove or disprove: If $A$ is a finite ring such that there exists a field $K,$ $K \subset A,$ then $|A|$ is a power of $|K|.$
0
votes
1answer
19 views

Series of Extension Fields

Let $ (a_n)_{n \in \mathbb{Z\geq0}} $ with $a_0 = 2 $ and $a_{n+1}= \sqrt{a_n}$ with $a_{n+1}>0$. We need to show that $[\mathbb{Q}(a_n):\mathbb{Q}]=2^n $$\forall n \in \mathbb{Q}$. To prove this ...
2
votes
2answers
29 views

Question about a Simple Field Extensions Equality

Let $E\supseteq F$ be an extension of fields. Show that $\forall u \in E,$ and nonzero $a\in F,$ $F(u)=F(au)$. My first instinct was to argue with the fact that $F(u)$ is the smallest subfield that ...
1
vote
1answer
67 views

Show that $U(A)\cup\{0\}$ is a field and $|A|\equiv1\pmod p$

Let $p$ be a prime number and $A$ a finite ring in which the group $U(A)$ of the invertible elements has order $p$. If there is an element $a\in U(A)$ such that $1-a\in U(A)$, show that $U(A)\cup\{0\}$...
0
votes
0answers
19 views

Computing the sum of inverses of some roots of 1 in a field, given their sum

Fix an algebraically closed field $F$. Let $\alpha_1,\dotsc,\alpha_n\in F$ be roots of $1$. Let $x=\alpha_1+\dotsc+\alpha_n$ and $y=\alpha_1^{-1}+\dotsc+\alpha_n^{-1}$. I was thinking: Given $n$ and ...
1
vote
0answers
25 views

Understanding the defining polynomial of the finite field $F_{p^d}$

I came across the following in my Cryptography notes: Definition: Let $p$ denote a prime number, and let $F_p$ denote the field with $p$ elements. Let $f(x)$ denote an irreducible polynomial over $...
0
votes
2answers
37 views

Factoring a polynomial in $\mathbb{Z_3}$ into linear factors.

I am trying to factor the polynomial $x^{7} - x$ over the field $\mathbb{Z_3}$. The solution is: $$x^{7} - x = x(x^6 - 1) = x(x^3 - 1)(x^3 + 1) = x(x - 1)^3(x + 1)^3.$$ I understand that the last ...