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Old 2017-10-04, 15:03   #12
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The question of determining the Hilbert Class Field was quite instructive to me in the use of Pari-GP. As a result, I can give defining polynomials for three quadratic extensions of the field k of 29th roots of unity, whose join is the Hilbert Class Field k(1) of k. The extension k(1)/k is elementary Abelian of degree 8; that is, the Galois group is the direct product of three cyclic groups of order 2.


x^2 + Mod(-3*t^27 - t^26 - 2*t^24 - 2*t^23 - t^21 - 3*t^20 - t^18 - t^17 - t^16 - t^15 - 2*t^14 - 2*t^11 - t^10 - t^8 - 2*t^7 - 2*t^4 - t^3 - t^2 - t - 1, t^28 + t^27 + t^26 + t^25 + t^24 + t^23 + t^22 + t^21 + t^20 + t^19 + t^18 + t^17 + t^16 + t^15 + t^14 + t^13 + t^12 + t^11 + t^10 + t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)

x^2 + Mod(-2*t^27 - 2*t^25 + t^24 - t^23 + t^22 - 2*t^21 - 2*t^19 - t^17 - t^16 - t^15 - 2*t^14 - t^13 - t^12 + t^11 - t^10 - t^9 - t^8 - t^7 + t^6 - t^5 - t^4 - 2*t^3 - t^2 - t - 1, t^28 + t^27 + t^26 + t^25 + t^24 + t^23 + t^22 + t^21 + t^20 + t^19 + t^18 + t^17 + t^16 + t^15 + t^14 + t^13 + t^12 + t^11 + t^10 + t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)

x^2 + Mod(t^27 + t^26 + 2*t^25 + t^24 + t^23 + t^21 + t^20 + t^19 + t^17 + t^16 + 2*t^15 + t^14 + t^13 + t^11 + t^10 + 3*t^9 + 2*t^8 + 2*t^7 + 2*t^4 + 2*t^3 + 3*t^2 + t + 1, t^28 + t^27 + t^26 + t^25 + t^24 + t^23 + t^22 + t^21 + t^20 + t^19 + t^18 + t^17 + t^16 + t^15 + t^14 + t^13 + t^12 + t^11 + t^10 + t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)

It was not difficult to find a degree-8 polynomial for k(1)/k. From it, I did produce a degree-224 polynomial in Z[x] defining k(1) as an extension of Q, but due to the high degree and large coefficients, I was unable to do any computations with it.
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Old 2017-10-05, 03:41   #13
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Okay, Thank you for the effort in finding these? Also I would like to know what the degree 8 polynomial in k the field of the 29th roots of unity, that defines the Hilbert Class Field L of k? Also, what is this polynomials discriminant?
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Old 2017-10-05, 04:17   #14
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A second announcement I would like to present, is the "creation" of new ideal classes in a certain field.

So suppose we take the field k of the 23rd roots of unity, which has class number 3. The primes p in k which are norms of principal ideals are the list in my previous three posts. In my previous two posts, I commented on the fact that in k, a prime p is a norm of a principal ideal if and only if there are solutions to x^3-x-1 = 0 (mod p). However the statement that in k, p is principal if and only if there are solutions to x^3-x-1 = 0 (mod p) is false because p = 6533 = 47*139 is a norm of a principal ideal in the field k of the 23rd roots of unity, but NOT in k = Q(sqrt(-23)). Let primes p, 'p, and q = 1 (mod 23). If P = p*'p, P is principal if and only if p and 'p are both not norms of principal ideals, or both are norms of principal ideals. P*q is always principal.

(20:59) gp > normU(23,x^3-x-1,1,100000)
Up to 300000 there are 429 primes congruent to 1 mod k and 141 are norms of principal ideals
(20:59) gp > print(v)
(20:59) gp>

We also have elements of the form a*x^21+b*x^20+c*x^19....t*x^2+u*x+v

for instance any element has a norm in k

(21:07) gp > norm(Mod(x^3+x^2-2*x-1,polcyclo(23)))
%1613 = 121356143

I.e. all the elements in k are mapped to a norm of a principal ideal in k, the field of the 23rd roots of unity. For each prime p a norm of principal ideal, there are exactly 22 elements which correspond to norm of p.

Now, to "create" a different ideal class, we will define a prime p = 1 (mod 23) as principal if and only if there are solutions to x^3-3*x-1 = 0 (mod p), and a prime with no solutions as non-principal. Again, the condition on P, is still true.

(20:59) gp > normU(23,x^3-3*x-1,1,100000)
Up to 300000 there are 429 primes congruent to 1 mod k and 136 are norms of principal ideals
(20:59) gp > print(v)
[]

Although x^3-3*x-1 does not define the Hilbert Class Field L of k, in the construction of this new "non-real" cyclotomic field c, it does.

In c, 47*139 = 6533 is still a norm of a principal ideal, and there should exist the same number of elements as in k. However, 599*829 = 496571 is not a norm of a principal in c and there are no elements with norm 496571.

The thing I am unsure about is using the new definition for a norm of a principal ideal in c, how do we compute the norm of an element in c?

For instance what elements w in c correspond to

norm(Mod(w,c)) = 1657

Since 1657 is prime, there should be exactly 22 elements in c whose norm is 1657. (Note 1657 is NOT principal in k, and there are no elements in the real field of the 23rd roots of unity.)

Last thing, which number field does c define? It is indeed a cyclotomic field of order 23, however the ideal classes are different compared to those in k.

Thanks for reading this, anyone into number fields, norms, elements, creation of different number fields, please help. Thanks.

Last fiddled with by carpetpool on 2017-12-11 at 06:00
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Old 2017-10-05, 04:53   #15
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Originally Posted by carpetpool View Post
A second announcement I would like to present, is the "creation" of new ideal classes in a certain field.

So suppose we take the field k of the 23rd roots of unity, which has class number 3. The primes p in k which are norms of principal ideals are the list in my previous three posts. In my previous two posts, I commented on the fact that in k, a prime p is a norm of a principal ideal if and only if there are solutions to x^3-x-1 = 0 (mod p). However the statement that in k, p is principal if and only if there are solutions to x^3-x-1 = 0 (mod p) is false because p = 6533 = 47*139 is a norm of a principal ideal in the field k of the 23rd roots of unity, but NOT in k = Q(sqrt(-23)). Let primes p, 'p, and q = 1 (mod 23). If P = p*'p, P is principal if and only if p and 'p are both not norms of principal ideals, or both are norms of principal ideals. P*q is always principal.

(20:59) gp > normU(23,x^3-x-1,1,100000)
Up to 300000 there are 429 primes congruent to 1 mod k and 141 are norms of principal ideals
(20:59) gp > print(v)
[]
(20:59) gp>

We also have elements of the form a*x^21+b*x^20+c*x^19....t*x^2+u*x+v

for instance any element has a norm in k

(21:07) gp > norm(Mod(x^3+x^2-2*x-1,polcyclo(23)))
%1613 = 121356143

I.e. all the elements in k are mapped to a norm of a principal ideal in k, the field of the 23rd roots of unity. For each prime p a norm of principal ideal, there are exactly 22 elements which correspond to norm of p.

Now, to "create" a different ideal class, we will define a prime p = 1 (mod 23) as principal if and only if there are solutions to x^3-3*x-1 = 0 (mod p), and a prime with no solutions as non-principal. Again, the condition on P, is still true.

(20:59) gp > normU(23,x^3-3*x-1,1,100000)
Up to 300000 there are 429 primes congruent to 1 mod k and 136 are norms of principal ideals
(20:59) gp > print(v)
[]

Although x^3-3*x-1 does not define the Hilbert Class Field L of k, in the construction of this new "non-real" cyclotomic field c, it does.

In c, 47*139 = 6533 is still a norm of a principal ideal, and there should exist the same number of elements as in k. However, 599*829 = 496571 is not a norm of a principal in c and there are no elements with norm 496571.

The thing I am unsure about is using the new definition for a norm of a principal ideal in c, how do we compute the norm of an element in c?

For instance what elements w in c correspond to

norm(Mod(w,c)) = 1657

Since 1657 is prime, there should be exactly 22 elements in c whose norm is 1657. (Note 1657 is NOT principal in k, and there are no elements in the real field of the 23rd roots of unity.)

Last thing, which number field does c define? It is indeed a cyclotomic field of order 23, however the ideal classes are different compared to those in k.

Thanks for reading this, anyone into number fields, norms, elements, creation of different number fields, please help. Thanks.
To help better understand this, consider everything you know about the field k of the 23rd roots of unity. This includes which numbers (and primes) are norms of principal ideals, which elements correspond to a norm of p, how many elements there are whose norm is p, compute the norm of an element in k, which polynomials define the Hilbert Class Field L of k, class number, etc. For the cyclotomic field c, everything is the same as k, except the polynomial that defines the Hilbert Class Field L of k, and maybe the class number.

In k, the polynomial is x^3-x-1 which defines the Hilbert Class Field L of k,
In c, the polynomial is x^3-3*x-1 which defines the Hilbert Class Field L of k,

For example, in k the element x-2 has norm 2^23-1 = 8388607 = 47*178481. In c what is the norm of x-2?

Not all numbers which are norms of principal ideals in k are the same in c, the ideal classes are different in both fields, but produce the same number of solutions.

Last fiddled with by carpetpool on 2017-12-11 at 06:01
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Old 2017-10-05, 15:48   #16
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Originally Posted by carpetpool View Post
So suppose we take the field k of the 23rd roots of unity, which has class number 3. The primes p in k which are norms of principal ideals are the list in my previous three posts. In my previous two posts, I commented on the fact that in k, a prime p is a norm of a principal ideal if and only if there are solutions to x^3-x-1 = 0 (mod p). However the statement that in k, p is principal if and only if there are solutions to x^3-x-1 = 0 (mod p) is false because p = 6533 = 47*139 is a norm of a principal ideal in the field k of the 23rd roots of unity, but NOT in k = Q(sqrt(-23)).
This is nonsense. If something is the norm of x in K/Q (a finite-degree extension of the rational field), it is automatically the norm from k/Q, of the norm from K to k of x, for any field k intermediate between K and Q.
(BTW, you're using the same symbol, k, for two different fields here.)

As to the solutions of the cubic mod p, I note that this only pertains to primes, and 47*139 isn't prime.
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Old 2017-10-06, 01:55   #17
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This is nonsense. If something is the norm of x in K/Q (a finite-degree extension of the rational field), it is automatically the norm from k/Q, of the norm from K to k of x, for any field k intermediate between K and Q.
(BTW, you're using the same symbol, k, for two different fields here.)

As to the solutions of the cubic mod p, I note that this only pertains to primes, and 47*139 isn't prime.
Okay, I might've gone too far on explaining my ideas, however I was referring to field extentions. The field K of nth roots of unity for prime n > 19, has a class number h > 1. There should be a field extension K/Q of the nth roots of unity which has class number h = 1 for ALL n. This also means ALL primes p congruent to 1 (mod n) are norms of principal ideals for infinitely many field extensions of the nth roots of unity K/Q.

Argument made simple, in the 23rd roots of unity, we see that 2*23+1 = 47 is the norm of an ideal, which is NOT principal, and no principal ideals have a norm of 47. There should exist infinitely many field extensions K/Q of the 23rd root of unity where 47 is the norm of a prinicpal ideal.

How to compute norms of elements in a field extension, I am unsure of, and also finding all elements with norm p in a rational field extension of the nth roots of unity.
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Old 2017-10-06, 14:35   #18
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The field K of nth roots of unity for prime n > 19, has a class number h > 1. There should be a field extension K/Q of the nth roots of unity which has class number h = 1 for ALL n.
Nope. It is well known that a number field can be embedded in a number field of class number 1 precisely when its Hilbert class field tower terminates. This is defined as follows. Given a number field K, let K(1) be its Hilbert class field, K(2) the Hilbert class field of K(1), K(3) the Hilbert class field of K(2), and so on. If it happens that K(i) has class number 1 for some i, then K(i+1) = K(i), and the tower terminates. It is well known that if L is a number field of class number 1 which contains K, then L contains K(i).

Alas, in many cases, the Hilbert class field tower does not terminate. In particular (see INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS, which cites

René Schoof, Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209–220),

this is the case for Q(\zeta_{877}).

In fact, it is true for "almost all" cyclotomic fields, as shown in

INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS
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Old 2017-10-07, 15:03   #19
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René's homepage (with many of his papers) can be found here: http://www.mat.uniroma2.it/~schoof/
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Old 2017-10-08, 12:13   #20
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Originally Posted by Dr Sardonicus View Post
Nope. It is well known that a number field can be embedded in a number field of class number 1 precisely when its Hilbert class field tower terminates. This is defined as follows. Given a number field K, let K(1) be its Hilbert class field, K(2) the Hilbert class field of K(1), K(3) the Hilbert class field of K(2), and so on. If it happens that K(i) has class number 1 for some i, then K(i+1) = K(i), and the tower terminates. It is well known that if L is a number field of class number 1 which contains K, then L contains K(i).

Alas, in many cases, the Hilbert class field tower does not terminate. In particular (see INFINITE HILBERT CLASS FIELD TOWERS FROM GALOIS REPRESENTATIONS, which cites

René Schoof, Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209–220),

this is the case for Q(\zeta_{877}).

In fact, it is true for "almost all" cyclotomic fields, as shown in

INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS
From reading these two pdfs, I am unsure of the following answers to my questions below.

Okay, so is Q(\zeta_{n}), n = 877 the first case of a cyclotomic field not having its Hilbert class field terminate (i.e. counter example)?

Second, The field K of nth roots of unity for prime n > 19, has a class number h > 1. There should be a field extension K/Q of the nth roots of unity which there is an element (in that field extension) with norm of p for any prime p = 1 (mod n). Does this stand true?

Last, can the processes described here of computing the norm of an element in a cyclotomic field be applied to computing the norm of an element in a field extension?
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Old 2017-10-08, 14:13   #21
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From reading these two pdfs, I am unsure of the following answers to my questions below.

Okay, so is Q(\zeta_{n}), n = 877 the first case of a cyclotomic field not having its Hilbert class field terminate (i.e. counter example)?
I have no idea whether 877 is the smallest n for which Q(\zeta_n) has an infinite Hilbert class field tower. It is merely the smallest value that turned up in a quick search. For the purpose of addressing the question you asked, it isn't important -- the fact is, for almost all n, the kind of extension of Q(\zeta_{n}) you want simply does not exist.

Quote:
Second, The field K of nth roots of unity for prime n > 19, has a class number h > 1. There should be a field extension K/Q of the nth roots of unity which there is an element (in that field extension) with norm of p for any prime p = 1 (mod n). Does this stand true?
Here, you are using K/Q to denote two different things -- a field of n-th roots of unity, and an extension of that field. If K is the field of n-th roots of unity, the usual notation for an extension of K is something/K, say L/K. And no, there doesn't have to be anything (particularly an algebraic integer) in L of norm p -- if you mean the norm to Q. If there's an algebraic integer in L whose norm to Q is p, then its norm to K is an algebraic integer in K whose norm to Q is p -- that is, a principal generator of an integral ideal of norm p. If the prime ideal factors of pOK are non-principal, you're out of luck. I have previously given an example in which there were no elements in K -- integral or not -- of norm p.

If you mean L/K with an element from L to K of norm p, of course there is. If L is the extension of K obtained by adjoining an n-th root r of p (where K/Q is the field of n-th roots of unity, and n is greater than 2), then the norm from L to K of r is p.

Quote:
Last, can the processes described here of computing the norm of an element in a cyclotomic field be applied to computing the norm of an element in a field extension?
As to computing the norm, any of the definitions in the article seem fine. For an extension L/K, use the behavior of the norm in towers as described in the article to compute the norm from L to Q.

Last fiddled with by Dr Sardonicus on 2017-10-08 at 14:15
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Old 2017-10-08, 19:45   #22
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If you mean L/K with an element from L to K of norm p, of course there is. If L is the extension of K obtained by adjoining an n-th root r of p (where K/Q is the field of n-th roots of unity, and n is greater than 2), then the norm from L to K of r is p.
Oops, slight oopsadaisy

I shouldn't have excluded the value n = 2. If n is even, you need an n-th root of -p rather than of p. This also assumes x^n + p or x^n - p is irreducible over the field of n-th roots of unity. Alas, that isn't always true (e.g. p = 2, n = 8).

If n is even, any extension defined by polynomial with constant term p which is irreducible in K[x] will fill the bill. if n is odd, an irreducible polynomial with constant term -p instead of p will do the job.
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