# Tag Info

## Hot answers tagged roots-of-unity

27 votes
Accepted

### Evaluating a remarkable term for primes p = 5 (mod. 8)

According to Theorem 5.1 of http://math.mit.edu/~rstan/pubs/pubfiles/35.pdf, the number is $$\frac 1p\left[ 2^{p-1}+\frac{p-1}{2}(\epsilon^{4h} + \epsilon^{-4h})\right],$$ where $h$ is the ...
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19 votes

### How small can a sum of a few roots of unity be?

This question grabbed my attention a couple of years ago and I've just put a paper on the arXiv with new upper bounds for $k=5$. I began by computing lots of data, then teased out the structure of ...
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15 votes
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8 votes

### $q$ as a prime power and a root of unity

This is very not rigorous, but it's a way of thinking about this topic which I find personally helpful. Several $\mathbb F_1$ papers contains remarks about the idea going back to Weil and Iwasawa that ...
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8 votes
Accepted

### Why are most coefficients of these minimal polynomials divisible by $p$?

So $\zeta=-\xi$, where $\xi$ is a primitive $p$'th root of unity. Let $\pi=1-\xi$. The minimal polynomial of $(1+\zeta^n)a$ is a product of terms of the form $X-(1+\zeta^{nj})\alpha_{ij}$, where $j$ ...
8 votes

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4 votes
Accepted

### Has any one seen this sum of roots of unity before?

Using the sagemath code, ...
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3 votes

### A conjecture involving roots of unity

Motivated by Nemo's solution in the case $\delta=0$, here I provide a proof for the case $\delta=1$. Let $\zeta$ be a primitive $m(n-1)+1$-th root of unity, and consider S=\sum_{k=1}^{n-1}\left(\...
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In the ring $\mathbb{Z}[\omega_p]$, the OP's second sum $\sum_{k=1}^{p-1} \left(\frac{k^2+k}{p}\right) \omega_p^{kl}$ raised to the $p$-th power is congruent to $\sum_{k=1}^{p-2} \left(\frac{k^2+k}{p}... • 88.3k 3 votes Accepted ### How to calculate$N_{L/k}$(roots of unity)? The answer should be: if$n = m$, then the image is the roots of unity of order$p^{n-1}$. If$n = m - 1$then the norm map should be surjective on the roots of unity. In the first case, since all ... • 1,413 2 votes ### Unit in cyclotomic field This is only a partial answer: it reduces the question to another problem. Taking up Ilya Bogdanov's comment, let$a$be a root of$2x^2 + x + 2\$. Then (as he says) we are interested in the set of ...
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