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Let p >5 be a prime number. Prove that every algebraic integer of the $p$th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.

Reference: http://www.artofproblemsolving.com/Forum/resources.php?c=2&cid=152&year=1977&sid=151602f87027a7ce87d3aa9421a666e9 Question No: 4

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    @Pete: I think it's quite reasonable to pose the question that I did above. If indeed the answer is far beyond the OPs knowledge then that information certainly plays a large role in deciding how to give an optimal answer. The OP has a history of rapidly posing diverse little-motivated questions from problem-books, so we should especially encourage him to provide some motivation and background.2010-08-25

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Miklos Schweitzer is a very hard contest.

Anyway, solution for this (and other problems) can be found in the book:

Contests in Higher Mathematics, published by Springer.

Google books has it:

http://books.google.com/books?id=2wwXImJ2HocC

And this particular problem's solution appears here:

http://books.google.com/books?id=2wwXImJ2HocC&pg=PA88

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    for some reason I cannot view this book.2010-08-24