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July 18

Polynomial whose roots are powers of another polynomial

Hello again. This is part of a theorem I am reading on single Hensel lift of a polynomial. Suppose I am given a polynomial in h_m in $Z_{p^{m}}[x]$ which divides x^k-1 as well. By Hensel's lemma I can construct a polynomial h(x) which is equivalent to it mod p^m and which divides x^k-1 too but in mod p^m+1. If we let x to be the root of h_m and y=x+p^mi the root of h then we have x^k=1+p^me as h_m divides x^k-1 in $Z_{p^{m}}[x]$ . Also in $Z_{p^{m+1}}[x]$ ; y^p=x^p and y^kp=1. All this is fine. My book now claims (and so does the link) that if I consider another polynomial h_m+1 whose roots are the pth powers of the roots of h then these roots coincide mod p^m with those of h_m. I can't justify this statement. I'll be grateful for any help. Thanks--Shahab (talk) 10:17, 18 July 2009 (UTC)[reply]

Irreducibility of multivariate polynomials

(redirect from sci desk) I cannot find anything about that. I can think of two reasons for this: either it's utterly complicated and not well understood or it can be so easily reduced to the question of irreducibility of polynomials of a single variable that no one cares to tell. What is it? 93.132.138.254 (talk) 09:35, 18 July 2009 (UTC) —Preceding unsigned comment added by 83.100.250.79 (talk) [reply]

Start maybe from here : Algebraic geometry.... --pma (talk) 11:47, 18 July 2009 (UTC)[reply]

Ah, your're right! I already have guessed this had something to do with mathematics somehow! 93.132.138.254 (talk) 14:44, 18 July 2009 (UTC)[reply]

My understanding was that Algebraic geometry primarily studied polynomials over algebraically-closed fields. What about polynomials over general rings? Can you provide any particular insight there? Eric. 76.21.115.76 (talk) 21:30, 18 July 2009 (UTC)[reply]

Not sure what kind of insights you are after, but as a starting point, multivariate polynomials over any unique factorization domain form themselves a UFD because of Gauss' lemma. — Emil J. 12:44, 20 July 2009 (UTC)[reply]

How does this work?

How does this work. I recently had it emailed to me and it seems to work. Thanks. Computerjoe 's talk 22:33, 18 July 2009 (UTC)[reply]

Warning: link plays unsolicited audio. Algebraist 22:34, 18 July 2009 (UTC)[reply]

Let your initial number have first digit x and second y, so the number is 10x+y. Subtracting the sum of digits from this gives 9x. So all the page has to do is label all multiples of 9 with the same option and it wins. Algebraist 22:37, 18 July 2009 (UTC)[reply]

Ah okay. Thanks. Computerjoe 's talk 22:46, 18 July 2009 (UTC)[reply]

but in fact it's a bit more subtle. As soon as you get the arithmetic trick, you try to cheat with n≠9x, and it still gets it right... The fact is that it knows the square where you are pointing the mouse on, when you choose it...--pma (talk) 08:19, 19 July 2009 (UTC)[reply]

Ummm... No...

I have not noticed that behavior. --COVIZAPIBETEFOKY (talk) 14:23, 19 July 2009 (UTC)[reply]

oh i really almost thought it read my mind... now i understand! it changes every time. --pma (talk) 16:37, 19 July 2009 (UTC)[reply]