# Tagged Questions

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 ...

20 views

24 views

### Subring of a field [closed]

Let $R$ be a subring of a field $F$ such that for each $x\in F$ either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I\subseteq J$ or $J\subseteq I$.
33 views

### Expanding an expression in a certain field

If $\mathbb F_2$ is a field of characteristic $2$, then we have $x+x=y+y=z+z=0$ for all $x,y,z \in \mathbb F_2$. When I expand $(x+y)(y+z)(z+x)$, I get \begin{align} (x+y)(y+z)(z+x) &= ...
17 views

### Completion of surreal subfields

Let $\kappa$ be a regular uncountable ordinal. Let $No(\kappa)$ denote the field of surreal numbers of birthdate $< \kappa$. In Fields of surreal numbers and exponentiation (2000), P. Ehrlich and ...
59 views

### Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field?

A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is ...
19 views

### Degree of irreducible polynomial

Let $\mathbb{F}$ be algebraic closure of field $\mathbb{k}$ and $[\mathbb{F}:\mathbb{k}] = n < \infty$. I have proved that if $f$ is irreducible over $\mathbb{k}$ than $\deg f | n$. But if $d | n$ ...
57 views

### I can't understand the formal definition of $\mathbb{R}$

I've always intuitively understood this set in intuitive sense, as "all numbers on the number line". However, now I want to know the formal definition: Consider the set of rational numbers, ...
32 views

### Finite extensions of $\mathbb F_p(t)$ [on hold]

Let $\mathbb F_p(t)$ the field of rational functions with coefficients in $\mathbb F_p$. Is it true or not that every finite extension $K$ of $\mathbb F_p(t)$ is $K\cong\mathbb F_{p^m}(t)$ for some ...
32 views

### Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where ...
33 views

### The sum of all primitive $n$-th roots of unity in the algebraic closure of $F(x)$

If $n$ is any natural number, then $\mu(n)$ is the sum of all primitive $n$-th roots of unity in $\mathbb{C}$ where $\mu$ is the Möbius function defined as $\mu(n)=0$ if $n$ is not square free and ...
46 views

### Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable?

Let $E/F$ a field extension. If $x\in E$ is separable over $F$ is $E/F$ separable ? I would say yes since the fact that $x$ separable over $F$ implies $E(x)/F$ separable, an since $E=E(x)$ then $E/F$ ...