View Single Post
 2013-06-11, 22:14 #3 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 4E916 Posts Please clarify limited questions 1. Suppose that we need to factor ideal (6) in number field Z[√-5], (6) factors as (2, 1+√-5)² (21) splits as (3, 1+2√-5) × (3, 1-2√-5) I have got some doubts on to how cubic number fields work out off... Please show how ideal (43) factoring in number field Z[³√2]. 43 = 9³ - 2 × 7³, 2 is being a cubic residue (mod 43). so that 43 is of form a²+27b², with a = 4, b = 1. Does it work as a companion to a³-2b³ form representation? Can every prime p of the form a²+27b² can be even written as form a³-2b³ form? Please factor ideal (31) in number field Z[³√2] 2 is being a cubic residue (mod 31). such that 31 is of form a²+27b², with a = 2, b = 1. Is 31 being of form a³-2b³ form? Solving a³-2b³ = 31... equation 2. If I want to initiate a number field by using PARI/GP, nfinit() function, and then it will ask for a generating polynomial. What will be the generating polynomial for the Z[√-6] being? Is it being x²+6? What will be the generating polynomial for the Z[³√2] being? Is it being x³-2? What will be the difference between nfinit() function, bnfinit() function? Consider polynomial frequently being used in SNFS, Cunningham project x4-x3+x2-x+1, being used with some Cunningham number; exponent of some multiple of 5. x6+x5+x4+x3+x2+x+1, being used with some Cunningham number; exponent of some multiple of 7. x5+x4+4x3-3x2-3x+1; being used with some Cunningham number; exponent of some multiple of 11. x6-x5-5x4+4x3+6x2-3x-1; being used with some Cunningham number; exponent of some multiple of 13. What will be ring of integers being generated by using them? → Or alternatively , what does a number field being generated by using a following fixed polynomial mean? → and then that's it by using a following fixed PARI/GP, nfinit() function, 3. What do the ring of integers in Z[cbrt(2)] look like? Are they being p+q*cbrt(2)+r*cbrt(4)? p, q, r are being integers. 4. Prove that prime numbers p of form x^2+27*y^2 (discriminant = -108) are being for which 2 is being a cubic residue (mod p). Prove that prime numbers p of form x^2+32*y^2 (discriminant = -128) are being for which -4 is being an octic residue (mod p). Prove that prime numbers p of form x^2+64*y^2 (discriminant = -256) are being for which 2 is being a biquadratic residue (mod p). Prove that if prime number p ≡ 1 (mod 3), 3 is being a cubic residue (mod p), and then that 4*p is being of form x^2+243*y^2. Prove that prime numbers p of form 4*x^2+2*x*y+7*y^2 (discriminant = -108) are being for which 2 is being a cubic non-residue (mod p). Prove that prime numbers p of form 4*x^2+4*x*y+9*y^2 (discriminant = -128) are being for which -4 is being an octic non-residue (mod p). Prove that prime numbers p of form 4*x^2+4*x*y+17*y^2 (discriminant = -256) are being for which 2 is being a biquadratic non-residue (mod p).