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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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    Did you mean to leave off the p > 5 assumption which is in the problem you linked?2010-08-24
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    @Jason: No! I am sorry it was a mistake.2010-08-24
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    Why are you posting deep questions like this when your answers elsewhere [1] make it crystal clear that you don't even understand the most rudimentary number-theoretical concepts such as LCM? [1] http://math.stackexchange.com/questions/31182010-08-24
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    Let's try to keep our eye on the question and not the questioner. Whether a question is appropriate should depend very little on who is asking it.2010-08-24
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    @Pete: Please do tell how you propose to explain the answer to such a question to someone who has difficulties with rudimentary number theory concepts. **The level of knowledge of the OP is extremely relevant to providing a good answer.**2010-08-24
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    @BD: In this particular case, I would propose to answer as Moron did. His answer was accepted so, presumably, acceptable to the OP. Concerning the general issue you raise: I respectfully disagree but acknowledge that your position is also a reasonable one. If you want to continue this discussion, the meta site is the place for it.2010-08-25
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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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    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".)2010-08-24
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    And open book! No wonder I had trouble solving Miklos Schweitzer problems on AoPS...2010-08-24
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    for some reason I cannot view this book.2010-08-24