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wu :: forums    riddles    putnam exam (pure math) (Moderators: william wu, Grimbal, SMQ, Icarus, towr, Eigenray)    polynomial of degree n « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: polynomial of degree n  (Read 1123 times) Benny Uberpuzzler     Gender: Posts: 1024 polynomial of degree n   « on: Jan 28th, 2008, 8:25pm » Quote Modify If f(x) is a polynomial of degree n, prove that f(x)=0(mod p) has exactly n solutions if and only if f(x) divides x^p -x (mod p). IP Logged If we want to understand our world — or how to change it — we must first understand the rational choices that shape it. Obob Senior Riddler     Gender: Posts: 489 Re: polynomial of degree n   « Reply #1 on: Jan 29th, 2008, 1:28am » Quote Modify The main idea is to note that every element of Z/pZ is a root of xp-x IP Logged Benny Uberpuzzler     Gender: Posts: 1024 Re: polynomial of degree n   « Reply #2 on: Jan 29th, 2008, 1:40pm » Quote Modify Obob,   Well it's easy to show that if f(x) is a factor of x^p -x(mod p), of degree n, then f(x) has exactly n solutions, however I don't know how to prove the converse (ie if f(x) is of degree n and has n solutions then   f(x) divides x^p -x(mod p).   IP Logged If we want to understand our world — or how to change it — we must first understand the rational choices that shape it. Obob Senior Riddler     Gender: Posts: 489 Re: polynomial of degree n   « Reply #3 on: Jan 29th, 2008, 6:12pm » Quote Modify If say a is a root of f(x) (mod p), then x-a divides f(x) (mod p).  Prove this using the division algorithm for polynomials.  So f(x) factors as a product c(x-a1)...(x-an) for some number c, where the ai are the roots of f(x).  But by the same logic, xp-x factors as x(x-1)(x-2)...(x-(p-1)).  Hence f(x) divides xp-x. IP Logged Benny Uberpuzzler     Gender: Posts: 1024 Re: polynomial of degree n   « Reply #4 on: Jan 31st, 2008, 9:06pm » Quote Modify Oh yeah, I forgot that you could do things like that. Man, it wasn't really that difficult after all was it.   Thank you very much. IP Logged If we want to understand our world — or how to change it — we must first understand the rational choices that shape it. Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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