Questions about Number Fields

[Raman]

As for number fields: I do suggest onto studying them by using generating polynomial rather than ring of integers in Z[d^(1/k)]

(1) Consider ring of integers in Q[sqrt(-6)]. What's the polynomial generating this number field. Is
it x^2+6?
(2) Is the polynomial generating all set of Gaussian integers is being x^2+1?
(3) Does the set of numbers of form x*sqrt(-2)+y*sqrt(-3) really form a number field? x, y are being integers.
Apparently it is not being closed: as follows as: [a*sqrt(-2)+b*sqrt(-3)]*[c*sqrt(-2)+d*sqrt(-3)]
= -2ac-3bd+sqrt(-6)*(ad+bc).
(4) What are the number field elements being generated by using polynomial 2*x^2+3? Is it being isomorphic
with x^2+6? Or does it not generate with number field at all. What about 3*x^2+2 and 6*x^2+1? Which of them
are being isomorphic with each other? Or that one or more of these polynomials does not generate with a number field at all?
(5) Of number fields being generated by using following polynomials: x^2+5, 5*x^2+1, 2*x^2+2*x+3, 3*x^2+2*x+2
all of same discriminant value of -20, of course,
which of them are being isomorphic to each other; which of them do not generate a number field at all?
(6) What are ring of integers being generated by using following polynomial
as follows as: 2*x^2+2*x+3? Is it being a valid statement, first of all?

(7) 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.
Does their norm will have inherited multiplicative property -- i.e. product of norm of two integers in Z[cbrt(2)] will also
be a norm of some integer in Z[cbrt(2)], as in quadratic fields ... ?
(a^2+k*b^2)(c^2+k*d^2)
= |a*d+k*b*c|^2 + k*|a*d-b*c|^2
= |a*d-k*b*c|^2 + k*|a*d+b*c|^2
What is its generating polynomial for this following number field being? Is it x^3-2?
(8) Ring of integers being generated by using following polynomial x^3-1 is always being necessarily of following form
p+q*omega+r*omega^2? p, q, r are being integers.
(9) How do I determine ring of integers in number field being generated by using polynomial x^4-x^3+x^2-x+1? How do I determine
which of number fields generated by using of what other polynomials of same degree -- (in this case -- 4) are being isomorphic onto it?
(10) Consider number fields being generated by using some special polynomials cases as follows as:
x^5+x^4+4*x^3-3*x^2-3*x+1; being used with some Cunningham number; exponent of some multiple of 11.
x^6-x^5-5*x^4+4*x^3+6*x^2-3*x-1; being used with some Cunningham number; exponent of some multiple of 13.
How do I determine ring of integers in number field being generated by using it?
(11) Are there being some type of number fields being generated by using some of the other functions, e.g. for instance
trigonometric / logarithmic / exponential / hyperbolic functions, besides that of algebraic / polynomial functions alone?
(12) 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).

I need on to know a few things on to

as follows as

-- How do following ideals (7), (13), (19), (31), (37), (43) do factor on to ring of integers being generated by using following number field Q[cbrt(2)]?
-- What's the case with following ideals (5), (11), (17), (23), (29), (41), (47)?
-- How many generator elements for ideals do cubic fields require at most?
-- Does two generator elements for ideals always suffice for the quadratic fields?
-- What's the case with number fields of generating polynomial of degree n?
-- What's the class number of ring of integers being generated by using following number field Q[cbrt(2)]?
-- How do I determine hidden companion polynomials for certain given number fields being generated by using cubic polynomials; that are being inequivalent on to each other... of same discriminant even!
-- Those which are equivalent polynomials on to given number fields being generated by using cubic polynomials can be obtained by using linear transformation of given variables...
-- What do ring of integers being generated by using following number field Q[cbrt(2)] look like?
-- How do I determine even discriminant of given number fields being generated by using cubic polynomials?
-- What will be generating polynomial for the ring of integers being generated by using following cubic field Q[cbrt(2)] / underlying number field Q[cbrt(2)]? Is it being x^3-2?

[Raman]

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).

[Raman]

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).

[Raman]

As for number fields: I do suggest onto studying them by using generating polynomial rather than ring of integers in Z[d^(1/k)]

(1) Consider ring of integers in Q[sqrt(-6)]. What's the polynomial generating this number field. Is
it x^2+6?
(2) Is the polynomial generating all set of Gaussian integers is being x^2+1?
(3) Does the set of numbers of form x*sqrt(-2)+y*sqrt(-3) really form a number field? x, y are being integers.
Apparently it is not being closed: as follows as: [a*sqrt(-2)+b*sqrt(-3)]*[c*sqrt(-2)+d*sqrt(-3)]
= -2ac-3bd+sqrt(-6)*(ad+bc).
(4) What are the number field elements being generated by using polynomial 2*x^2+3? Is it being isomorphic
with x^2+6? Or does it not generate with number field at all. What about 3*x^2+2 and 6*x^2+1? Which of them
are being isomorphic with each other? Or that one or more of these polynomials does not generate with a number field at all?
(5) Of number fields being generated by using following polynomials: x^2+5, 5*x^2+1, 2*x^2+2*x+3, 3*x^2+2*x+2
all of same discriminant value of -20, of course,
which of them are being isomorphic to each other; which of them do not generate a number field at all?
(6) What are ring of integers being generated by using following polynomial
as follows as: 2*x^2+2*x+3? Is it being a valid statement, first of all?

(7) 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.
Does their norm will have inherited multiplicative property -- i.e. product of norm of two integers in Z[cbrt(2)] will also
be a norm of some integer in Z[cbrt(2)], as in quadratic fields ... ?
(a^2+k*b^2)(c^2+k*d^2)
= |a*d+k*b*c|^2 + k*|a*d-b*c|^2
= |a*d-k*b*c|^2 + k*|a*d+b*c|^2
What is its generating polynomial for this following number field being? Is it x^3-2?
(8) Ring of integers being generated by using following polynomial x^3-1 is always being necessarily of following form
p+q*omega+r*omega^2? p, q, r are being integers.
(9) How do I determine ring of integers in number field being generated by using polynomial x^4-x^3+x^2-x+1? How do I determine
which of number fields generated by using of what other polynomials of same degree -- (in this case -- 4) are being isomorphic onto it?
(10) Consider number fields being generated by using some special polynomials cases as follows as:
x^5+x^4+4*x^3-3*x^2-3*x+1; being used with some Cunningham number; exponent of some multiple of 11.
x^6-x^5-5*x^4+4*x^3+6*x^2-3*x-1; being used with some Cunningham number; exponent of some multiple of 13.
How do I determine ring of integers in number field being generated by using it?
(11) Are there being some type of number fields being generated by using some of the other functions, e.g. for instance
trigonometric / logarithmic / exponential / hyperbolic functions, besides that of algebraic / polynomial functions alone?
(12) 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).

I need on to know a few things on to

as follows as

-- How do following ideals (7), (13), (19), (31), (37), (43) do factor on to ring of integers being generated by using following number field Q[cbrt(2)]?
-- What's the case with following ideals (5), (11), (17), (23), (29), (41), (47)?
-- How many generator elements for ideals do cubic fields require at most?
-- Does two generator elements for ideals always suffice for the quadratic fields?
-- What's the case with number fields of generating polynomial of degree n?
-- What's the class number of ring of integers being generated by using following number field Q[cbrt(2)]?
-- How do I determine hidden companion polynomials for certain given number fields being generated by using cubic polynomials; that are being inequivalent on to each other... of same discriminant even!
-- Those which are equivalent polynomials on to given number fields being generated by using cubic polynomials can be obtained by using linear transformation of given variables...
-- What do ring of integers being generated by using following number field Q[cbrt(2)] look like?
-- How do I determine even discriminant of given number fields being generated by using cubic polynomials?
-- What will be generating polynomial for the ring of integers being generated by using following cubic field Q[cbrt(2)] / underlying number field Q[cbrt(2)]? Is it being x^3-2?

[R.D. Silverman]

Originally Posted by Raman

As for number fields: I do suggest onto studying them by using generating polynomial rather than ring of integers in Z[d^(1/k)]

(1) Consider ring of integers in Q[sqrt(-6)]. What's the polynomial generating this number field. Is
it x^2+6?
(2) Is the polynomial generating all set of Gaussian integers is being x^2+1?
(3) Does the set of numbers of form x*sqrt(-2)+y*sqrt(-3) really form a number field? x, y are being integers.
Apparently it is not being closed: as follows as: [a*sqrt(-2)+b*sqrt(-3)]*[c*sqrt(-2)+d*sqrt(-3)]
= -2ac-3bd+sqrt(-6)*(ad+bc).
(4) What are the number field elements being generated by using polynomial 2*x^2+3? Is it being isomorphic
with x^2+6? Or does it not generate with number field at all. What about 3*x^2+2 and 6*x^2+1? Which of them
are being isomorphic with each other? Or that one or more of these polynomials does not generate with a number field at all?
(5) Of number fields being generated by using following polynomials: x^2+5, 5*x^2+1, 2*x^2+2*x+3, 3*x^2+2*x+2
all of same discriminant value of -20, of course,
which of them are being isomorphic to each other; which of them do not generate a number field at all?
(6) What are ring of integers being generated by using following polynomial
as follows as: 2*x^2+2*x+3? Is it being a valid statement, first of all?

(7) 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.
Does their norm will have inherited multiplicative property -- i.e. product of norm of two integers in Z[cbrt(2)] will also
be a norm of some integer in Z[cbrt(2)], as in quadratic fields ... ?
(a^2+k*b^2)(c^2+k*d^2)
= |a*d+k*b*c|^2 + k*|a*d-b*c|^2
= |a*d-k*b*c|^2 + k*|a*d+b*c|^2
What is its generating polynomial for this following number field being? Is it x^3-2?
(8) Ring of integers being generated by using following polynomial x^3-1 is always being necessarily of following form
p+q*omega+r*omega^2? p, q, r are being integers.
(9) How do I determine ring of integers in number field being generated by using polynomial x^4-x^3+x^2-x+1? How do I determine
which of number fields generated by using of what other polynomials of same degree -- (in this case -- 4) are being isomorphic onto it?
(10) Consider number fields being generated by using some special polynomials cases as follows as:
x^5+x^4+4*x^3-3*x^2-3*x+1; being used with some Cunningham number; exponent of some multiple of 11.
x^6-x^5-5*x^4+4*x^3+6*x^2-3*x-1; being used with some Cunningham number; exponent of some multiple of 13.
How do I determine ring of integers in number field being generated by using it?
(11) Are there being some type of number fields being generated by using some of the other functions, e.g. for instance
trigonometric / logarithmic / exponential / hyperbolic functions, besides that of algebraic / polynomial functions alone?
(12) 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).

I need on to know a few things on to

as follows as

-- How do following ideals (7), (13), (19), (31), (37), (43) do factor on to ring of integers being generated by using following number field Q[cbrt(2)]?
-- What's the case with following ideals (5), (11), (17), (23), (29), (41), (47)?
-- How many generator elements for ideals do cubic fields require at most?
-- Does two generator elements for ideals always suffice for the quadratic fields?
-- What's the case with number fields of generating polynomial of degree n?
-- What's the class number of ring of integers being generated by using following number field Q[cbrt(2)]?
-- How do I determine hidden companion polynomials for certain given number fields being generated by using cubic polynomials; that are being inequivalent on to each other... of same discriminant even!
-- Those which are equivalent polynomials on to given number fields being generated by using cubic polynomials can be obtained by using linear transformation of given variables...
-- What do ring of integers being generated by using following number field Q[cbrt(2)] look like?
-- How do I determine even discriminant of given number fields being generated by using cubic polynomials?
-- What will be generating polynomial for the ring of integers being generated by using following cubic field Q[cbrt(2)] / underlying number field Q[cbrt(2)]? Is it being x^3-2?

Would someone please move these ramblings to the misc.math forum?
These questions can be easily answered if the poster would only bother
to pick up a book on elementary algebraic number theory (Larry Washington's
is excellent; so is Weiss's; stay away from Lang) and READ IT.

[R.D. Silverman]

Originally Posted by R.D. Silverman

Would someone please move these ramblings to the misc.math forum?
These questions can be easily answered if the poster would only bother
to pick up a book on elementary algebraic number theory (Larry Washington's
is excellent; so is Weiss's; stay away from Lang) and READ IT.

H. Cohen's book (Vol I & II) would also be excellent.