Discriminants and Large Pólya Groups in Septic Number Fields

Technical Summary: Discriminants and Large Pólya Groups in Septic Number Fields

Problem Statement
The systematic classification of Pólya fields and the understanding of the magnitude of Pólya groups across families of number fields remain central challenges in algebraic number theory. While significant progress has been made for low-degree extensions (quadratic, cubic, quartic, and sextic), the study of cyclic fields of degree 7 (septic) has lagged. Specifically, no infinite family of cyclic septic fields admitting a complete and explicit determination of their Pólya groups has previously appeared in the literature. Furthermore, the question of how large Pólya groups can become within such families, and whether one can construct blocks of consecutive fields with simultaneously large Pólya groups, remains largely unexplored for degree 7.

Methodology
The paper investigates a specific parametric family of cyclic septic fields $\{K_t\}_{t \in \mathbb{Z}}$ arising from the Hashimoto–Hoshi construction, defined by the roots of the polynomial:
$$f_t(X) = X^7 + a_6(t)X^6 + \dots + a_0(t)$$
where the coefficients are polynomials in $t$. The core of the methodology involves:

1. Explicit Discriminant Computation: Using the framework of Spearman and Williams for cyclic extensions, the author performs a Tschirnhausen transformation to eliminate the $X^6$ term, resulting in a transformed polynomial $f^*_t(X)$. By analyzing the divisibility properties of the coefficients of this transformed polynomial and applying Theorem 2.1, the discriminant $d(K_t)$ and the conductor $f(K_t)$ are computed explicitly.
2. Arithmetic of a Sextic Polynomial: The analysis reveals that the arithmetic of the Pólya groups in this family is governed entirely by the prime factorization of a single sextic polynomial:
   $$E(t) = t^6 + 2t^5 + 11t^4 + t^3 + 16t^2 + 4t + 8$$
   The conductor and discriminant are shown to be determined by the prime divisors of $E(t)$ that are congruent to $1 \pmod 7$.
3. Power-Free Conditions: The study assumes that $E(t)$ takes fifth-power free values. This condition is justified by Theorem 2.3 (Erdős), which guarantees infinitely many such integers, and Theorem 2.4, which provides asymptotic density results.
4. Group Theoretic Analysis: Utilizing Chabert's formula for the cardinality of Pólya groups in cyclic extensions and Zantema's characterization of ambiguous ideal classes, the structure of the Pólya group $Po(K_t)$ is derived in terms of the number of distinct prime divisors of $E(t)$, denoted $\omega(E(t))$.
5. Constructive Proofs for Consecutive Fields: To address the existence of consecutive fields with large Pólya groups, the author employs Schur's Lemma and the Chinese Remainder Theorem to construct blocks of integers $t, t+1, \dots, t+m-1$ such that the associated polynomials $E(t+i)$ possess a prescribed number of distinct prime factors.

Key Contributions and Results

• Explicit Characterization of Pólya Groups: The paper provides a complete characterization of the Pólya group for the Hashimoto–Hoshi family. Under the assumption that $E(t)$ is fifth-power free, the Pólya group is isomorphic to:

  • $(\mathbb{Z}/7\mathbb{Z})^{\omega(E(t))-2}$ if $t$ is even.
  • $(\mathbb{Z}/7\mathbb{Z})^{\omega(E(t))-1}$ if $t$ is odd.
    Consequently, $K_t$ is a Pólya field if and only if $\omega(E(t))$ equals 2 (for even $t$) or 1 (for odd $t$).

• Unboundedness of Pólya Groups: The family contains infinitely many non-Pólya fields for which the cardinality of the Pólya group is unbounded. Specifically, the 7-rank of the class groups of these fields is unbounded. This is established by showing that for a positive proportion of primes $p$, $E(7p+2)$ is fifth-power free, and by Theorem 2.5 (Halberstam), $\omega(E(7p+2))$ tends to infinity.

• Consecutive Large Pólya Groups: The paper proves that for any fixed integers $m \ge 2$ and $r > 1$, there exist infinitely many blocks of $m$ consecutive fields in the family such that the 7-rank of the Pólya group for each field in the block is at least $r$. This is achieved by constructing arithmetic progressions where the values of the associated polynomials are simultaneously divisible by distinct sets of primes.

• Monogeneity and Field Index: The paper demonstrates that for any odd integer $t$ coprime to 15, the field $K_t$ is non-monogenic. However, the field index $I(K_t)$ is exactly 1. This result is derived by showing that the discriminant of the defining polynomial is not divisible by primes less than the degree (2, 3, or 5) under these conditions, while invoking Gras's theorem which implies that a cyclic septic field cannot be monogenic unless it is the maximal real subfield of a specific cyclotomic field (which is impossible for degree 7).

Significance and Claims
The primary significance of this work lies in filling a gap in the literature regarding cyclic septic fields. It provides the first infinite family of such fields where the Pólya groups can be explicitly determined and shown to be arbitrarily large.

• Theoretical Implications: The results offer an alternative proof for the conjecture that the $m$-rank of the class group of a number field of degree $d$ is unbounded when $m=d=7$.
• Conditional Nature of Pólya Fields: The paper notes that while it proves the existence of infinitely many non-Pólya fields with large groups, the existence of infinitely many Pólya fields in this family is conditional. It relies on Bunyakovsky's conjecture, which posits that the irreducible polynomial $E(t)$ takes prime values infinitely often. The paper explicitly states that proving the infinitude of Pólya fields unconditionally would settle Bunyakovsky's conjecture for $E(t)$, a problem currently open for all polynomials of degree greater than one.
• Limitations: The authors are modest regarding the unconditional existence of Pólya fields in this family, acknowledging that the current results depend on the fifth-power free assumption and, for the existence of Pólya fields specifically, on the unproven Bunyakovsky conjecture. The paper does not propose new experimental applications but rather advances the theoretical understanding of the structure of Pólya groups in higher-degree cyclic extensions.