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

  • $\begingroup$ Did you mean to leave off the p > 5 assumption which is in the problem you linked? $\endgroup$ – Jason DeVito Aug 24 '10 at 1:35
  • $\begingroup$ @Jason: No! I am sorry it was a mistake. $\endgroup$ – anonymous Aug 24 '10 at 2:06
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    $\begingroup$ 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] math.stackexchange.com/questions/3118 $\endgroup$ – Bill Dubuque Aug 24 '10 at 3:17
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    $\begingroup$ 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. $\endgroup$ – Pete L. Clark Aug 24 '10 at 15:34
  • $\begingroup$ @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. $\endgroup$ – Bill Dubuque Aug 24 '10 at 23:24

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:


And this particular problem's solution appears here:


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    $\begingroup$ 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: en.wikipedia.org/wiki/Mikl%C3%B3s_Schweitzer_Competition. (It's a "take-home exam".) $\endgroup$ – Pete L. Clark Aug 24 '10 at 2:47
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    $\begingroup$ And open book! No wonder I had trouble solving Miklos Schweitzer problems on AoPS... $\endgroup$ – Qiaochu Yuan Aug 24 '10 at 2:54
  • $\begingroup$ for some reason I cannot view this book. $\endgroup$ – anon Aug 24 '10 at 8:44

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