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 December 16th, 2018, 08:05 PM #1 Newbie   Joined: Dec 2018 From: Earth Posts: 4 Thanks: 1 Hi everyone, I am new here. I have been trying to figure out how to attack multiple problems involving cyclotomic polynomials, roots of unity, number fields, and other number theory subjects related to those. One particular problem I am stuck on is the following: Let $n$ and $p$ be primes such that $p=1\pmod n$. Find the smallest set containing k distinct roots of unity $S=[\zeta_{n_1},\zeta_{n_2},\zeta_{n_3},\zeta_{n_4},.. .\zeta_{n_k}]$ such that $\zeta_{n} + \zeta_{n_2} + \zeta_{n_3} + \zeta_{n_4} ... + \zeta_{n_k} = 0 \pmod p$ Is there an algorithm for solving this problem for a particular n and p (especially if p is relatively large compared to n)? Estimating the behavior of this problem, one can predict that the value of $k$ is approximately $\log_n(p)$. It is also worth asking which sums are unique (because if $S$ is a solution set, then at least $n-2$ other solution sets exists, and therefore all the subset sums are not necessarily unique). Another way of stating this problem is finding the smallest $u = \zeta_{n} + \zeta_{n_2} + \zeta_{n_3} + \zeta_{n_4} ... + \zeta_{n_k}$ such that the norm $N$ of the element $u$ in the field $K=\mathbb Q(\zeta_{n})$ is divisible by $p$. This also resorts to asking the following question (which will most likely answer my original question): What is the smallest integer $N$ such that $N=0\pmod p$ and $N$ is the norm of an element $u$ in the field $K=\mathbb Q(\zeta_{n})$? In particular, how can this integer $N$ be found? Sorry if I have described the problem in broad manner. Here is an example of the problem where I fix $n$ and $p$. Example: $n=31, p=1117$ We see that $p=1\pmod n$, and $\zeta_{n} + \zeta_{n_2} + \zeta_{n_3} + \zeta_{n_4} + \zeta_{n_k} = 0 \pmod p$ where $\zeta_{n} + \zeta_{n}^2 + \zeta_{n}^3 + \zeta_{n}^8 + \zeta_{n}^{13} = 0 \pmod p$ as the norm of $\zeta_{n} + \zeta_{n}^2 + \zeta_{n}^3 + \zeta_{n}^8 + \zeta_{n}^{13}$ in $K=\mathbb Q(\zeta_{n})$ is $1455451 = 1117*1303 = 0 \pmod p$. However, the "smallest" $N$ that is divisible by $p$ is $N=139625=5^3*1117=0\pmod p$ which is generated by norm the element $u = -\zeta_{n} + \zeta_{n}^2 + \zeta_{n}^4$ (the norm of $u$ in $K$ is $N$). I am currently trying to attack this problem in general for any prime $n$ and a prime $p=1\pmod n$ under the conditions $p<2^{(n-1)/2}$ and $N<2^n$ (Indeed if $p<2^{(n-1)/2}$ holds, then $N<2^n$ should also hold). Any ideas? Will give thanks for any helpful responses. Again, sorry if the description of the problem I am trying to solve is too broad. Last edited by skipjack; December 17th, 2018 at 02:59 AM. Tags roots, sum, unity, vanishing Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post jiasyuen Algebra 3 February 3rd, 2015 08:36 AM not3bad Complex Analysis 2 December 20th, 2014 02:35 AM Jacob0793 Complex Analysis 3 March 7th, 2014 05:58 AM mathbalarka Number Theory 37 January 14th, 2014 04:57 AM Elladeas Abstract Algebra 2 February 19th, 2011 02:20 PM

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