# 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.

9,191 questions
Filter by
Sorted by
Tagged with
1answer
33 views

1answer
73 views

### Which field does $\overline{\mathbb{C}(t)}$ refer to?

I'm reading a textbook on algebraic geometry and it mentions the field $\overline{\mathbb{C}(t)}$ without defining it. It does define $\mathbb{C}(t)$ as the field of rational functions with ...
2answers
37 views

### Is there a difference between the Galois group of $K/F$ and the Galois group of $E/K$ where $K$ is an intermediate field?

Given an extension $E/F$, with intermediate fields $E/K_1/K_2/…../F$, I want to know if $Gal(K_n/F)=Gal(E/K_n)$. By Galois correspondence it seems like this should be true, there are still the same ...
1answer
22 views

0answers
64 views

### Can I think of a field as a distributive category?

My understanding is that a field $(k, +, \times)$ is a set $k$ with abelian group structures $(k, 0, +)$ and $(k \setminus \{0\}, 1, \times)$ such that multiplication distributes over addition. Can ...
3answers
190 views

### Extending Galois and Cyclotomic Fields

In this question I asked whether, given a Galois field $K$ with Galois group $H$, it was possible to find a Galois extension $L/K/\mathbb{Q}$ of a desired degree with Galois group $G$ and $H \leq G$ ...
0answers
38 views

### Find me a field with these properties. [duplicate]

So I am searching for a field $K$ with these properties: $1.$ $K$ is infinite. $2.$ $K$ is not algebraically closed. $3.$ $K$ is algebraic over a finite field. I do not think the answer will be ...
1answer
31 views

### Showing $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$. [duplicate]

Let $p$ and $q$ be prime numbers, $p\neq q$. Now I have to show $L:=\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})=:K$. The direction $K\subseteq L$ is clear. For the other ...
0answers
36 views

### Is the finite field $Z_p$ an ordered one? Or can a finite field be ordered? [duplicate]

Since residue classes modulo $p$ are not numbers but sets of numbers with several operations defined on them, i don't think we can compare them at all. Yet i'm not sure. For sake of an example lets ...
2answers
59 views

0answers
47 views

### Purely transcendental extensions do not affect the minimal polynomial.

I've been studying for qualiying exams and recently came across this problem. Let $L/M/K$ be a field extensions with $[L:M]<\infty$. Let $A$ be a subfield consisting of all elements of $L$ that ...
1answer
43 views

### How can we prove the existence of algebraic closures from first-order compactness?

A devilish remark in „Aluffi: Algebra Chapter 0“[1, Remark VII.2.7] claims that $^7$ Allegedly, the existence of algebraic closures is a consequence of the compactness theorem for first-odrer logic,...
0answers
53 views

### How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt[7]{5}, \exp(2\pi i/7))$?

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt[7]{5}$ and $\omega = \exp(2\pi i/7)$. Now, ...
2answers
63 views

### A faster way to find the quadratic extensions of a field extension?

The usual method I have for finding Galois correspondence goes like this : Say we have the Splitting field of $x^4-3$, i.e. $\Bbb Q(i,\sqrt[4]{3})$. Then it's generating automorphisms are those ...
0answers
39 views

1answer
26 views

### Union of subgroups of $\mathbb{C^*}$

Let $\mathbb{ℂ^∗}$ = $\mathbb{ℂ}$\ ${0}$ denote the group of non-zero complex numbers under multiplication. Suppose $Y_n$={$z\in \mathbb{C} |z^n=1$} then which of the following is (are) subgroups ...
1answer
20 views

### For $x$ lives in the algebraic closure of $\mathbb F_q$, $x^{q-1}=1$ if and only if $x \in \mathbb F_q$?

Let $\mathbb F_q$ be a finite field of order $q$ and let $C$ be its algebraic closure. Taking $x\in C$, I wonder if $x^{q-1}=1$ if and only if $x \in \mathbb F_q$. Of course $x \in \mathbb F_q$ ...
2answers
76 views

### Fermat's little theorem as a consequence of $(x+y)^p=x^p+y^p$ in $\Bbb Z_p$?

Exercise in a book I am currently reading asks that I prove Fermat's little theorem as a consequence of the fact that in $\Bbb Z_p$ for any $x$ and $y$, $(x+y)^p=x^p+y^p$. Although this feels like it ...
0answers
21 views

### Implications between two properties involving the ring $~K[x,y]~$

I tried to solve this problem, but I'm not sure if my solution is correct. The problem is the following one: Let $~K~$ be a field, $~f(x)~$ and $~g(x)~$ two irreducible polynomials in $~K[x]~$ with ...
1answer
788 views

### Proof that every field is perfect?

The following must be wrong, since it shows that every field is perfect, which I gather is not so. But I can't find the error: Suppose $E/K$ is a field extension and $p\in K[x]$ is irreducible ...
0answers
36 views

1answer
43 views

### Proving that multiplicative groups of two fields F* are isomorphic

Considering the fields that are described by: $$\mathbb{F}_i = \mathbb{Z}_2[x] /\langle\mkern 1.5mu p_i(x)\mkern1.5mu\rangle, \enspace i=1,2$$ where $p_1(x)=x^3 + x + 1$ and $p_2(x)=x^3+x^2+1$ I ...
0answers
32 views

### Is my line of reasoning correct for considering this fixed field as a simple extension over $GF(7)$?

Suppose we have the field of rational functions $GF(7)(t)$ in the indeterminate $t$ over $GF(7)$. Deﬁne $σ,τ ∈ Aut(E)$ by $σ(t) = 2t$ and $τ(t) = 1 /t$. Set $G = <σ,τ>$ and $F = Fix(G)$. There ...
1answer
21 views

### Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable?

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable ? I have a feeling that it has something to do with the idea that if $E:F$ is separable hence ...
1answer
39 views

### A quicker way to decide if this extension is Galois.

Given $\alpha=\sqrt{1+\sqrt{2}}$, a min. poly for this element over $\Bbb Q$ is $x^4-2x^2-1$, as it's monic irreducible over $\Bbb Q$ and has $\alpha$ as a root. The roots of this min. polynomial ...
1answer
52 views

### What theorem is being invoked in this solution to deciding if this polynomial is solvable by radicals?

I was reading these solutions online http://campus.lakeforest.edu/trevino/Spring2019/Math331/Homework7Solutions.pdf for practice for an upcoming exam I have . But in question $2$ part $(a)$ the author ...
0answers
47 views

### Is the largest order of an element of a group always the size of the group? [duplicate]

Similar to how one can find a primitive root element of $\mathbb{Z}/p\mathbb{Z}$, I was wondering if, for any group of size $N$ that can be written as $(k\backslash\{0\},\cdot)$ where $k$ is a field, ...