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Old 2013-06-11, 22:14   #3
Raman
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"Mr. Tuch"
Dec 2007
Chennai, India

3·419 Posts
Default 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).
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