 mersenneforum.org Polynomials defining the same field as cyclotomic polynomial order 5
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 Register FAQ Search Today's Posts Mark Forums Read 2017-04-19, 20:33 #1 carpetpool   "Sam" Nov 2016 5×67 Posts Polynomials defining the same field as cyclotomic polynomial order 5 I haven't been on here a while, I was studying further information and knowledge on monic polynomials P(x) = x^4 + a*x^3 + b*x^2 + c*x + d defining the same field as cyclotomic polynomial 5 C_5(x) = x^4 + x^3 + x^2 + x + 1. A conjecture I made a while back reverting from this states that there are infinitely many polynomials P(x) defining the same field as C_n(x) for fixing the largest (n+1)/2 coefficients (a_k) (for prime n) as long as they satisfy conditions a (mod n) = t^(n-2), a_2 (mod n) = t^(n-3), a_3 (mod n) = t^(n-4),...... a_((n-1)/2) = t^((n+1)/2)). This seems to be proven for C_5(x) and the proof for C_7(x) might require more knowledge about cubic polynomials and fields. Proof: Choose any two integers; A and B such that A^2 (mod 5) = B (mod 5). and solve -C^2 + D^2 + T^2 = (8*B-3*A^2)/5  If D + A + C + T = 2 (mod 4), (if not then change the sign of T so it is). then solve [D = a - b - c + d, A = a + b + c + d, C = - a - b + c + d, T = a - b + c - d]  with the solution set (a, b, c, d) write; a*x + b*x^2 + c*x^3 + d*x^4  then take the minimum polynomial of [a*x + b*x^2 + c*x^3 + d*x^4, x^4 + x^3 + x^2 + x + 1] Since we can choose infinitely many values of T, we will get infinitely many polynomials defining the same field as C_5(x). Example; take A = 9, B = 21, 9^2 (mod 5) = 21 (mod 5), -C^2 + D^2 + T^2 = (8*21-3*9^2)/5 = -15 choosing T = 1, D^2 - C^2 = 16 D = 5 C = 3 then we check 5 + 9 + 3 + 1 = 18 = 2 (mod 4). then solve [5 = a - b - c + d, 9 = a + b + c + d, 3 = - a - b + c + d, 1 = a - b + c - d] a = 3, b = 0, c = 2, d = 4 Replacing the coefficients into : 3*x + 0*x^2 + 2*x^3 + 4*x^4 = 3*x + 2*x^3 + 4*x^4 take the minimum polynomial of [3*x + 2*x^3 + 4*x^4, x^4 + x^3 + x^2 + x + 1] = x^4 + 9*x^3 + 21*x^2 + 19*x + 131 I am making efforts in researching the field of C_7(x) and finding a similar equation and system of solutions for fixing the first four coefficients (1, a, b, c) of x^6 + a*x^5 + b*x^4 + c*x^3 + d*x^2 + f*x + g. Fixing the coefficient of x^3 requires more knowledge about cubic forms, and I don't have the knowledge or theories for this. Anyone else care to investigate this problem? I spent a lot of time and effort trying to find and come up with these assertions. Thanks for help, comments, suggestions.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post carpetpool carpetpool 24 2017-10-29 23:47 carpetpool Miscellaneous Math 3 2017-04-07 02:15 carpetpool Miscellaneous Math 2 2017-02-25 00:50 paul0 Programming 6 2015-01-16 15:12 mickfrancis Factoring 2 2015-01-11 18:31

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