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0 votes
1 answer
223 views

On the Irreducibility of Cyclotomic polynomials

Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
2 votes
1 answer
158 views

Computing the minimal polynomial of roots of polynomials with algebraic coefficients

Let $p(x) = \sum_{i=0}^{n} c_i x^i$ with $c_i \in \mathbb{A}$ with $q_i(c_i) = 0$ and all $q_i \in \mathbb{Q}[x]$ being minimal polynomials of the coefficients. Let $r$ be a zero of $p(x)$. Is there ...
10 votes
1 answer
327 views

Proving that polynomials belonging to a certain family are reducible

In an article, I've found the following result. Unfortunately, it was derived from a general, somewhat complicated theory, that would be cumbersome for this result alone. Assume that $\mathbb F_p$ is ...
2 votes
0 answers
112 views

Abelian approximation of fields

Given a field extension $K/k$ of finite degree and a norm $d$ on $\overline{k}$, what is the smallest real number $\alpha_{d}^K$ such that for every element $z$ of $K$ there is an element $z^a$ of $k^...
3 votes
1 answer
136 views

Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals

Suppose $K=\mathbb{Q}(X_1,\dots,X_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be ...
2 votes
2 answers
291 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...