Articles of field theory

How to prove that $k(x)$ is not complete in the $x$-adic metric

It is not hard to find proofs showing that $\mathbb{Q}$ is not complete with respect to the metric induced by the valuation $|\;\;|_p$. For example, it is enough to recall that every complete metric space is Baire and to show that $(\mathbb{Q},|\;\;|_p)$ is not Baire. Another option is to show that $\mathbb{Q}_p$ is uncountable, so […]

Multiplicative Property of the degree of field extension

According to Artin’s Algebra, chapter 15, section 3, the mapping property of the degree of field extension is as follows: Let $F\subset K\subset L$ be fields. Then $[L:F]=[L:K][K:F]$, where $[K:F]$ represents the dimension of $K$, as an $F$-vector space. My question is based on its Corollary: Let $\mathcal{K}$ be an extension field of a field […]

What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$

We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R \times \mathbb R$ into a field non-isomorphic to this usual field ?

BIG Intermediate fields in an infinite tower

This question arose out of the answer by Brandon Carter to the following question: Infinite dimensional intermediate subfields of an algebraic extension of an algebraic number field Define an infinite tower of field extensions recursively, starting from an algebraic number field $K=K_1$ in this way: That is [$K_{n+1}:K_n]=d_n, d_{n}>1$ with $K_{n+1}$ obtained from $K_n$ by […]

Can an algebraic extension of an uncountable field be of uncountable degree?

I want to show that, if $K$ is a maximal subfield of $\mathbb C$ without $\sqrt{2}$ in it, then $\mathbb C$ is of countable degree over $K$. Is it the case that an algebraic extension of an uncountable field always has countable degree? if so, I can try to show $K$ is uncountable.

Finding a primitive element of a finite field

Let $F = \mathbb{F}_p[x]/(m(x))$, where $m(x)$ is irreducible in $\mathbb{F}_p[x]$. How do I find a primitive element of $F$, i.e., one that generates the nonzero elements of $F$ multiplicatively? For example, in $\mathbb{F}_{256} = \mathbb{F}_2[x]/(x^8+x^4+x^3+x+1)$, the element $x+1$ has order $255$, so it is a primitive element of $\mathbb{F}_{256}$.

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the equation$$\text{N}(a) = 1$$where now we take $a \in \mathbb{Q}(i)$. What is a method for producing solutions to this equation using the arithmetic […]

Finding Galois group of $x^6 – 3x^3 + 2$

I’m trying to find the Galois group of $$f(x)= x^6 – 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 – 2)$$ I can see the splitting field must be $\mathbb{Q}(\omega, \sqrt[3]{2})$ (where $\omega$ is a 3rd root of unity) which has degree $6$ and so […]

The product map and the inverse map are continuous with respect to the Krull topology

Let $K/F$ be a Galois extension with $G=\text{Gal}(K/F)$. $\mathcal{I}=\lbrace E\text{ }|\text{ }E/F \text{ is Galois and }[E:F]<\infty\rbrace$ $\mathcal{N}=\lbrace N\text{ }|\text{ } N=\text{Gal}(K/E) \text{ for some } E\in\mathcal{I}\rbrace$ The Krull topology over $G$ is defined as follows. The set $B=\lbrace \sigma_iN_i\text{ }|\text{ }\sigma_i\in G \text{ and }N_i\in\mathcal{N}\rbrace$ is the base for the Krull topology. I am […]

Question on the normal closure of a field extension

Hi all I was given the following question in my field theory class which I am stuck on: I am given $ K/F $ a finite extension of fields. I am asked to show the existence of an $ K \subset L $ such that L/F is a normal extension and if we have $ […]