Tagged Questions

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Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both ...
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General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset. Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...
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Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it: Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...
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Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in ... 2answers 518 views trace zero elements in algebraic number fields If we are given an algebraic number field L, and$ \alpha $is an element of L whose field trace over Q is zero and whose field norm over Q belongs to Z, then does$ \alpha $necessarily belong to the ... 3answers 2k views What is the current status of Agrawal's conjecture? In their famous 'Primes is in P' paper Agrawal, Kayal and Saxena stated the following conjecture: If for coprime integers$n$and$r$the equality$(X-1)^n = X^n - 1$holds in ... 0answers 253 views Looking for product of symmetric polynomials evaluated at roots of unity Consider$a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$where$\alpha$is a complex$N$th root of unity where$N = 2 + ...
This may seems to be an elementary question, but I found no answers on MO nor google. I have always heard "polynomials are easier to handle with than integers". For example: When $n$ is quite ...
I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem: Let $A (n \times n)$ be a ...