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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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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

Aryabhata
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    when I followed your link and looked at the problems in this undergraduate competition, I thought: "What the heck? I'm a research mathematician and I feel lucky to understand the statements of these problems. Undergraduates are asked to solve them on the spot?!?" So I googled and found this, which allowed me to pick up the pieces of my exploded skull and more or less glue them back together: http://en.wikipedia.org/wiki/Mikl%C3%B3s_Schweitzer_Competition. (It's a "take-home exam".) – Pete L. Clark Aug 24 '10 at 02:47
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    And open book! No wonder I had trouble solving Miklos Schweitzer problems on AoPS... – Qiaochu Yuan Aug 24 '10 at 02:54
  • for some reason I cannot view this book. –  Aug 24 '10 at 08:44