# 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.

8,723 questions
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### 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 $\mathbb Q$...
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### 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. ...
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### 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 $F \simeq \Bbb Q[x]/\langle f(x) \rangle$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 "...
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### 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 $f$ ...
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### 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.
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### 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 ...
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### (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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb Z_n$ ...
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### 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.
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### 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 ...
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### 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.
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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 $(R,... 1answer 31 views ### Show that$\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$containing$\...
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 $|K|.$