Tagged Questions
4
votes
2answers
76 views
Radical extension
Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$.
It's ...
0
votes
1answer
67 views
Roots of unity in a field generated by a root of a polynomial
The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root
k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ...
1
vote
2answers
100 views
show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity
Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that
$1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$.
My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?
0
votes
1answer
112 views
Cyclotomic Polynomials
Let $E(n)$ denote an nth root of unity. (For convenience, we may take $E(n) = \exp(\frac{2πi}{n})$.)
Prove that for any prime $p$ and any natural number $r$, we have
$$
\prod_{\substack{j\\ \gcd(p^r, ...
2
votes
1answer
104 views
Fields modulo $n$-th powers, discrete valuations and roots of unity
I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with ...
