(a) Show that, for any odd prime p > 5, G P = 1 if p...

 

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(a) Show that, for any odd prime p > 5, G P = 1 if p 1,-5 (mod 12) -1 if p = -1,5 (mod 12) (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 + 3 has roots in Z/61Z by (a): they are +27 +61Z (you do not need to verify this). Using this information, give the roots of the polynomial X² - X + 19 in Z/61Z. (a) Show that, for any odd prime p > 5, G P = 1 if p 1,-5 (mod 12) -1 if p = -1,5 (mod 12) (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 + 3 has roots in Z/61Z by (a): they are +27 +61Z (you do not need to verify this). Using this information, give the roots of the polynomial X² - X + 19 in Z/61Z.

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Guiven P5 and P is a odd prime P is not congruent to 2468... View the full answer