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).
Numbers integrally represented by a ternary 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.
