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2
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0answers
151 views

Cyclotomic integers with given modulus

The following problem was posted to the NMBRTHRY mailing list about a week ago, without eventually getting a satisfactory solution. Suppose that $p=(n^2+1)/2$ is a prime, with $n\ge 5$ integer. Does ...
0
votes
1answer
387 views

When does the modulus of a sum of an integer and an algebraic integer equal an integer?

Let say Z is a sum of n-roots of unity and thus an algebraic integer, and D is an rational integer. If |z+D| is an integer, what can we conclude regarding Z? can we say |Z| is integer? Another ...
14
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3answers
3k views

Quick proof of the fact that the ring of integers of Q(\zeta_n) is Z[\zeta_n]?

I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of ...
2
votes
1answer
330 views

is there any bound on the absolute number of algebraic integer in terms of its degree?

If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))? Clearly we need to have distinct roots of unity otherwise this ...
8
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3answers
745 views

which algebraic integers in a cyclotomic field give you integer absolute value?

Does anyone know an answer to this question? Question: In an cyclotomic field which algebraic integers have integer absolute value? Revision 1: -1 I like to add this to the above question, Let's ...
7
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3answers
1k views

Ring of algebraic integers in a quadratic extension of a cyclotomic field

Hello, I have a question which arose when trying to classify orders of certain algebras. We know that if $K=\mathbb{Q}(\zeta)$ is any cyclotomic field, and $\zeta$ is an $n$-th root of unity (for ...
7
votes
2answers
617 views

How can I prove that a sequence of squares of graph norms is never cyclotomic?

The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G. Now, fix some graph ...