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, 12√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 a³2b³ form representation? Can every prime p of the form a²+27b² can be even written as form a³2b³ 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 a³2b³ form?
Solving a³2b³ = 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 x³2?
What will be the difference between nfinit() function,
bnfinit() function?
Consider polynomial frequently being used in SNFS, Cunningham project
x^{4}x^{3}+x^{2}x+1, being used with some Cunningham number; exponent of some multiple of 5.
x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1, being used with some Cunningham number; exponent of some multiple of 7.
x^{5}+x^{4}+4x^{3}3x^{2}3x+1; being used with some Cunningham number; exponent of some multiple of 11.
x^{6}x^{5}5x^{4}+4x^{3}+6x^{2}3x1; 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 nonresidue (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 nonresidue (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 nonresidue (mod p).
