4
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In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but not $r^2$ or $r,$ and that these representations should be predictable based on the "decomposition type" when factoring the original one-variable polynomial $\pmod p.$ I should probably emphasize that I am not requiring the ability to separate the primes by congruences. That does happen for the quartic example $x^4 + x^3 + x^2 + x + 1,$ but not for the two cubic examples I've asked about.

I just spoke with Ken Ribet, showed him the Lenstra example below, how badly the values of $f(x,y,z,t)$ come out, and asked whether he knew of some lists of examples that worked well (he didn't). The examples I know are the principal form of binary quadratic forms, especially one class per genus but definitely including some others, as positive forms with class number 3 or 4(articles by Williams).

https://math.stackexchange.com/questions/2284356/what-numbers-are-integrally-represented-by-this-quartic-polynomial-norm-form

Numbers integrally represented by a ternary cubic form

https://math.stackexchange.com/questions/329936/primes-represented-integrally-by-a-homogeneous-cubic-form

I had this idea that reducibility $\pmod p$ would mean that the homogeneous polynomial constructed, the norm form, would integrally represent low powers of that prime, especially $p$ or $p^2$ or $p^3,$ in agreement with the "decomposition type" of the original polynomial.

I have since gotten feedback that this (or something like it) is the degenerate case, specified in Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? As a result of the answer by Franz, I have requested the Frohlich book on central extensions and Franz's book on reciprocity laws.

This is from the polynomial example $x^4 + 3 x^2 + 7x + 4$ discussed in the framework of Chebotarev Density in Lenstra-Stevenhagen, especially pages 10,11,12 in this pdf. The article appeared in 1996 in The Mathematical Intelligencer.

I guess I should repeat that they point out that $x^4 + 3 x^2 + 7x + 4$ is reducible mod every prime. This has thrown me off, I had this idea that reducibility would mean that the homogeneous polynomial below, the norm form, would integrally represent low powers of that prime, especially $p$ or $p^2$ or $p^3.$ Something very much like that happened with these earlier questions,

I thought I had something reasonable which is to take the companion matrix $$ A = \left( \begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -4 & -7 & -3 & 0 \end{array} \right) $$ and create $$ f(x,y,z,t) = \det \left( x I + y A + z A^2 + t A^3 \right) $$ $$ f(x,y,z,t) = x^4 + (-6z - 21t)x^3 + (3y^2 + (21z - 2t)y + (17z^2 + 21tz + 138t^2))x^2 + (-7y^3 + (-16z + 42t)y^2 + (-21z^2 - 99tz - 203t^2)y + (25z^3 + 28tz^2 + 139t^2z - 91t^3))x + (4y^4 - 24ty^3 + (12z^2 + 84tz + 68t^2)y^2 + (-28z^3 - 64tz^2 - 84t^2z + 100t^3)y + (16z^4 + 48t^2z^2 - 112t^3z + 64t^4)) $$

Today I also borrowed A Course in Computational Algebraic Number Theory by Henri Cohen; Introduction to the Construction of Class Fields by Cohn; The Genus Fields of Algebraic Number Fields by Ishida; Algebraic Number Theory by Frohlich; Algebraic Number Fields by Janusz.

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  • 1
    $\begingroup$ Can't you just take any field of class number one? For example, $x^6-x-1$. Then either $p^6$, $p^3$, $p^2$, or $p$ will be the smallest power of $p$ which is a norm, depending on whether the polynomial splits as follows: degree $6$ in the first case, degree $3+3$ in the second, degrees $2+4, 2+2+2$ in the third, or anything else for the last. $\endgroup$ – Pound Sterling Jul 19 '17 at 15:39
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I did some calculations for the example $x^6 - x - 1$ as in comment by Pound Sterling. Long computer run; it does appear to work as advertised. In particular, when $x^6 - x - 1$ factors with decomposition type 3+3, I was able to find representations of that prime cubed. It would appear that that is the statement for class number one, decomposition type says what (smallest) power of the prime is represented.

I was able to finish the first half dozen primes that give cubes, this output in gp-pari:


? K = bnfinit(x^6 - x - 1);
? K.disc 
%10 = 49781
? factor(K.disc)
%11 = 
[67 1]
[743 1]

? K.clgp
%12 = [1, [], []] 
===========================================================================
// 157, 241, 283, 347, 421, 577, 607, 677, 1087, 1117, 1181, 1361, 1511, 1523
------------------------------------------------------------
? factormod(x^6 - x - 1, 157)
[Mod(1, 157)*x^3 + Mod(63, 157)*x^2 + Mod(133, 157)*x + Mod(87, 157) 1]
[Mod(1, 157)*x^3 + Mod(94, 157)*x^2 + Mod(68, 157)*x + Mod(83, 157) 1]

? id = [ 1,0,0,0,0,0; 0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1]
%1 = 
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]

? a = [ 0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; 1,1,0,0,0,0]
%2 = 
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
[1 1 0 0 0 0]

? charpoly(a)
%3 = x^6 - x - 1
? matdet(a)
%4 = -1
? h = 6 * id - 5 * a - 4 * a^2 + 6 * a^3 + 1 * a^4 + 5 * a^5  
%5 = 
[6 -5 -4 6 1 5]
[5 11 -5 -4 6 1]
[1 6 11 -5 -4 6]
[6 7 6 11 -5 -4]
[-4 2 7 6 11 -5]
[-5 -9 2 7 6 11]
? matdet(h)
%6 = 3869893
? factor(matdet(h))
%7 = 
[157 3]
=================================================================================
? factormod(x^6 - x - 1, 241)
%10 = 
[Mod(1, 241)*x^3 + Mod(2, 241)*x^2 + Mod(111, 241)*x + Mod(201, 241) 1]
[Mod(1, 241)*x^3 + Mod(239, 241)*x^2 + Mod(134, 241)*x + Mod(235, 241) 1]

? h = 2 *  id +  5 * a - 11 * a^2 - 6 * a^3 - 9 * a^4 + 0 * a^5
%3 = 
[2 5 -11 -6 -9 0]
[0 2 5 -11 -6 -9]
[-9 -9 2 5 -11 -6]
[-6 -15 -9 2 5 -11]
[-11 -17 -15 -9 2 5]
[5 -6 -17 -15 -9 2]

? matdet(h)
%4 = 13997521
? factor(matdet(h))
%5 = 
[241 3]
============================================================================
? factormod(x^6 - x - 1, 283)
%9 = 
[Mod(1, 283)*x^3 + Mod(85, 283)*x^2 + Mod(30, 283)*x + Mod(25, 283) 1]
[Mod(1, 283)*x^3 + Mod(198, 283)*x^2 + Mod(120, 283)*x + Mod(249, 283) 1]

? h = 8 * id - 4 * a + 9 * a^2 - 9 * a^3 + 3 * a^4 + 4 * a^5  
%6 = 
[8 -4 9 -9 3 4]
[4 12 -4 9 -9 3]
[3 7 12 -4 9 -9]
[-9 -6 7 12 -4 9]
[9 0 -6 7 12 -4]
[-4 5 0 -6 7 12]

? matdet(h)
%7 = 22665187
? factor(matdet(h))
%8 = 
[283 3]
=============================================================================
? factormod(x^6 - x - 1, 347)
%8 = 
[Mod(1, 347)*x^3 + Mod(8, 347)*x^2 + Mod(3, 347)*x + Mod(18, 347) 1]
[Mod(1, 347)*x^3 + Mod(339, 347)*x^2 + Mod(61, 347)*x + Mod(212, 347) 1]

? h = 11 * id + 4 * a + 8 * a^2 + 15 * a^3 + 13 * a^4 + 10 * a^5 
%4 = 
[11 4 8 15 13 10]
[10 21 4 8 15 13]
[13 23 21 4 8 15]
[15 28 23 21 4 8]
[8 23 28 23 21 4]
[4 12 23 28 23 21]

? matdet(h)
%5 = 41781923
? factor(matdet(h))
%6 = 
[347 3]
============================================================================
? factormod(x^6 - x - 1, 421)
%7 = 
[Mod(1, 421)*x^3 + Mod(119, 421)*x^2 + Mod(45, 421)*x + Mod(31, 421) 1]
[Mod(1, 421)*x^3 + Mod(302, 421)*x^2 + Mod(223, 421)*x + Mod(258, 421) 1]

? h = 13 * id - 4 * a + 12 * a^2 + 11 * a^3 + 6 * a^4 + 3 * a^5 
%4 = 
[13 -4 12 11 6 3]
[3 16 -4 12 11 6]
[6 9 16 -4 12 11]
[11 17 9 16 -4 12]
[12 23 17 9 16 -4]
[-4 8 23 17 9 16]

? matdet(h)
%5 = 74618461
? factor(matdet(h))
%6 = 
[421 3]
=========================================================================
? factormod(x^6 - x - 1, 577)
%1 = 
[Mod(1, 577)*x^3 + Mod(76, 577)*x^2 + Mod(473, 577)*x + Mod(387, 577) 1]
[Mod(1, 577)*x^3 + Mod(501, 577)*x^2 + Mod(110, 577)*x + Mod(82, 577) 1]

? h = 13 * id - 14 * a - 10 * a^2 + 0 * a^3 + 2 * a^4  + 11 * a^5 
%8 = 
[13 -14 -10 0 2 11]
[11 24 -14 -10 0 2]
[2 13 24 -14 -10 0]
[0 2 13 24 -14 -10]
[-10 -10 2 13 24 -14]
[-14 -24 -10 2 13 24]

? matdet(h)
%9 = 192100033
? factor(matdet(h))
%10 = 
[577 3]
===============================================================
? factormod(x^6 - x - 1, 607)
%1 = 
[Mod(1, 607)*x^3 + Mod(284, 607)*x^2 + Mod(425, 607)*x + Mod(535, 607) 1]
[Mod(1, 607)*x^3 + Mod(323, 607)*x^2 + Mod(107, 607)*x + Mod(548, 607) 1]
=========================================================================
? factormod(x^6 - x - 1, 677)
%2 = 
[Mod(1, 677)*x^3 + Mod(8, 677)*x^2 + Mod(329, 677)*x + Mod(185, 677) 1]
[Mod(1, 677)*x^3 + Mod(669, 677)*x^2 + Mod(412, 677)*x + Mod(505, 677) 1]
==========================================================

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