Partial Vandiver Theorem: Cyclotomic Insights

• Partial Vandiver theorem is a family of results that impose restricted splitting conditions in cyclotomic fields using specific cyclotomic-unit congruences.
• It employs Kummer extensions and residue-symbol techniques to derive criteria that are equivalent to or support Vandiver’s conjecture in SFLT2 and FLT2 frameworks.
• The theorem influences class group decompositions by establishing concrete conditions on p-principal primes and cyclotomic units in both number field and function-field settings.

“Partial Vandiver theorem” designates, in the cited literature, not a single universally fixed theorem but a family of Vandiver-type results that stop short of proving the full Kummer–Vandiver conjecture $p \nmid h_p^+$. In the cyclotomic setting $K=\mathbb{Q}(\zeta_p)$, these results typically assert restricted principality, total splitting in Kummer $p$-extensions generated by cyclotomic units, or explicit congruences for special units such as $1-\zeta$, under hypotheses tied to hypothetical counterexamples to the second case of Fermat’s Last Theorem or to the second case of the Strong Fermat’s Last Theorem conjecture $SFLT2$. Closely related uses of the term occur in Gauss-sum criteria for Vandiver’s conjecture, in Kummer-theoretic decompositions, and in function-field analogues where only restricted Herbrand–Ribet type statements survive (Quême, 2013, Queme, 2011, Gras, 2018, Anglès et al., 2011).

1. Classical framework and the restricted meaning of Vandiver-type results

Vandiver’s conjecture is the assertion that for every odd prime $p$,

$p \nmid h_p^+,$

where $h_p^+$ is the class number of the maximal real subfield

$K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$

of the $p$-th cyclotomic field $K=\mathbb{Q}(\zeta_p)$0. Equivalently, the $K=\mathbb{Q}(\zeta_p)$1-part of the class group of $K=\mathbb{Q}(\zeta_p)$2 lies entirely in the minus part, or, in the notation of the K-theoretic formulation, $K=\mathbb{Q}(\zeta_p)$3 (Quême, 2013, Stolin, 2020).

A central auxiliary notion is that of a $K=\mathbb{Q}(\zeta_p)$4-principal prime. For a rational prime $K=\mathbb{Q}(\zeta_p)$5, this means that for every prime ideal $K=\mathbb{Q}(\zeta_p)$6 of $K=\mathbb{Q}(\zeta_p)$7 above $K=\mathbb{Q}(\zeta_p)$8, the class $K=\mathbb{Q}(\zeta_p)$9 is a $p$0-power in the class group; equivalently,

$p$1

for some ideal $p$2 and $p$3. Under Vandiver’s conjecture, many primes $p$4 with suitable splitting conditions are automatically $p$5-principal. This is one of the main mechanisms by which full Vandiver information is converted into restricted local or decomposition-theoretic statements (Quême, 2013).

In the papers on $p$6 and FLT2, a “partial Vandiver theorem” therefore means a theorem that assumes only enough class-group input to control primes or units relevant to the Diophantine problem. The restriction is usually one of three kinds. It may concern only a specific family of primes, only a specific character component of a class group, or only a specific Kummer extension generated by cyclotomic units. This usage is explicit in Quême’s work, where the results are described as imposing “Vandiver-type conditions” on primes arising from $p$7 or FLT2 configurations (Queme, 2011).

A related but distinct analogue appears over function fields. There the naive Kummer–Vandiver analogue asks whether

$p$8

Taelman’s Herbrand–Ribet theorem gives control in the range $p$9, but Anglès, Ngo Dac, and Tavares Ribeiro construct counterexamples to the full analogue. In that setting, only restricted, hence partial, Vandiver-type statements remain valid (Anglès et al., 2011).

2. Partial Vandiver theorems in the $1-\zeta$0 setting

The second case of the Strong Fermat’s Last Theorem conjecture is formulated by the Diophantine equation

$1-\zeta$1

with $1-\zeta$2, $1-\zeta$3, $1-\zeta$4, and the second case condition $1-\zeta$5. In the cyclotomic field $1-\zeta$6, the element

$1-\zeta$7

is then a $1-\zeta$8-primary pseudo-unit if $1-\zeta$9 fails. This class-field-theoretic reformulation is the basis of the modern Vandiver-type approach (Quême, 2013).

Let $SFLT2$0 be an odd prime with $SFLT2$1, let $SFLT2$2 be the order of $SFLT2$3, let $SFLT2$4 be the order of $SFLT2$5, let $SFLT2$6 be a primitive $SFLT2$7-th root of unity, and let

$SFLT2$8

If $SFLT2$9 is $p$0-principal and $p$1, the main cyclotomic-unit congruence proved in this context is

$p$2

Equivalently, the $p$3-th power residue symbol of $p$4 at primes above $p$5 is trivial. The proof uses the $p$6-Hilbert class field, the fact that $p$7 is a $p$8-primary pseudo-unit, and the complete splitting of $p$9-principal primes in the relevant $p \nmid h_p^+,$0-Hilbert extensions (Quême, 2013).

This is called partial Vandiver-type because it does not prove Vandiver’s conjecture. Instead, it shows that if Vandiver’s predicted supply of $p \nmid h_p^+,$1-principal primes is available, then any hypothetical $p \nmid h_p^+,$2 counterexample must satisfy very rigid congruences in Kummer extensions generated by cyclotomic units. The paper further formulates a criterion, described there roughly as follows: if there are infinitely many primes $p \nmid h_p^+,$3 for which no prime above $p \nmid h_p^+,$4 satisfies the required cyclotomic-unit congruence, then $p \nmid h_p^+,$5 must hold for $p \nmid h_p^+,$6. This is a partial Vandiver theorem in the precise sense that failure of the expected Vandiver-type splitting pattern forces the Diophantine statement (Quême, 2013).

The same framework yields heuristic consequences. For large $p \nmid h_p^+,$7, assuming $p \nmid h_p^+,$8 fails and Vandiver’s conjecture holds, many primes $p \nmid h_p^+,$9 of even order modulo $h_p^+$0 should be $h_p^+$1-principal. Since total splitting in the Kummer extension cut out by the units $h_p^+$2 is heuristically rare, the paper argues that most such primes must divide $h_p^+$3. This suggests that a hypothetical counterexample would force $h_p^+$4 to have an unrealistically large number of prime divisors (Quême, 2013).

3. Vandiver’s cyclotomic integers and total splitting in Kummer extensions

A more explicit “partial Vandiver theorem” language appears in the companion work on Vandiver’s cyclotomic integers. There Quême introduces “Vandiver units”

$h_p^+$5

with $h_p^+$6, and also the totally real cyclotomic units

$h_p^+$7

These are the units used to encode decomposition laws via $h_p^+$8-th power residue symbols, in direct continuity with Vandiver’s classical method (Queme, 2011).

Assume $h_p^+$9 fails for $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$0, with $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$1, and let $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$2 be a $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$3-principal prime. With $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$4, $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$5, $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$6, and $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$7 as above, Lemma 2.2 shows that for all $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$8 and all primes $K^+ = \mathbb{Q}(\zeta_p+\zeta_p^{-1})$9 in $p$0,

$p$1

The corresponding decomposition theorem is that every $p$2 splits totally in

$p$3

This is the core Vandiver-style decomposition principle in the paper: under failure of $p$4, $p$5-principal auxiliary primes must split totally in highly nontrivial Kummer $p$6-extensions generated by Vandiver units (Queme, 2011).

The partial nature of the result is explicit. It applies only to primes tied to a putative $p$7 solution and only to the Kummer extensions generated by the chosen cyclotomic integers. The paper then formulates a stronger conjecture: for every pair $p$8 with $p$9, $K=\mathbb{Q}(\zeta_p)$00, and $K=\mathbb{Q}(\zeta_p)$01, there exists at least one $K=\mathbb{Q}(\zeta_p)$02-principal prime $K=\mathbb{Q}(\zeta_p)$03 such that the prime $K=\mathbb{Q}(\zeta_p)$04 does not split totally in the above extension. Since the theorem gives total splitting under $K=\mathbb{Q}(\zeta_p)$05 failure, this conjecture would imply $K=\mathbb{Q}(\zeta_p)$06 (Queme, 2011).

A second, weaker Vandiver-type conjecture in the same paper concerns the reduced form

$K=\mathbb{Q}(\zeta_p)$07

For irregular $K=\mathbb{Q}(\zeta_p)$08, Quême defines a finite set $K=\mathbb{Q}(\zeta_p)$09 of smallest non $K=\mathbb{Q}(\zeta_p)$10-principal primes whose classes generate $K=\mathbb{Q}(\zeta_p)$11, and proves that if $K=\mathbb{Q}(\zeta_p)$12, then there exists $K=\mathbb{Q}(\zeta_p)$13 with $K=\mathbb{Q}(\zeta_p)$14 such that primes above $K=\mathbb{Q}(\zeta_p)$15 split totally in a refined Kummer extension generated by modified products of the $K=\mathbb{Q}(\zeta_p)$16. The resulting criterion states that if such total splittings do not occur, then any $K=\mathbb{Q}(\zeta_p)$17 solution must satisfy $K=\mathbb{Q}(\zeta_p)$18 (Queme, 2011).

4. FLT2, Furtwängler-type refinements, and explicit congruences

The cyclotomic-unit method also produces FLT2-specific partial Vandiver theorems. In the $K=\mathbb{Q}(\zeta_p)$19 paper, if $K=\mathbb{Q}(\zeta_p)$20 divides $K=\mathbb{Q}(\zeta_p)$21 and is $K=\mathbb{Q}(\zeta_p)$22-principal, one obtains

$K=\mathbb{Q}(\zeta_p)$23

and

$K=\mathbb{Q}(\zeta_p)$24

If $K=\mathbb{Q}(\zeta_p)$25 divides $K=\mathbb{Q}(\zeta_p)$26 and is $K=\mathbb{Q}(\zeta_p)$27-principal, one obtains

$K=\mathbb{Q}(\zeta_p)$28

and

$K=\mathbb{Q}(\zeta_p)$29

These congruences are presented as strong refinements of the classical first and second theorems of Furtwängler, because they do not only recover congruences of the form $K=\mathbb{Q}(\zeta_p)$30, but also prescribe the local behavior of the cyclotomic unit $K=\mathbb{Q}(\zeta_p)$31 (Quême, 2013).

The exceptional case $K=\mathbb{Q}(\zeta_p)$32 clarifies the method. In the general theorem one requires $K=\mathbb{Q}(\zeta_p)$33, because when $K=\mathbb{Q}(\zeta_p)$34 one has $K=\mathbb{Q}(\zeta_p)$35 and some of the factors $K=\mathbb{Q}(\zeta_p)$36 vanish. The paper therefore treats this case separately and derives the congruence involving $K=\mathbb{Q}(\zeta_p)$37 directly. This specialized replacement is still of Vandiver type, but it is no longer expressed through the general ratios $K=\mathbb{Q}(\zeta_p)$38 (Quême, 2013).

In the FLT2-focused sequel, Quême proves further restricted principality statements. If FLT2 fails for $K=\mathbb{Q}(\zeta_p)$39 with $K=\mathbb{Q}(\zeta_p)$40, then under Vandiver’s conjecture every prime dividing

$K=\mathbb{Q}(\zeta_p)$41

must be $K=\mathbb{Q}(\zeta_p)$42-principal. For primes dividing

$K=\mathbb{Q}(\zeta_p)$43

the paper derives residue-symbol identities for the units $K=\mathbb{Q}(\zeta_p)$44, and when $K=\mathbb{Q}(\zeta_p)$45 it obtains

$K=\mathbb{Q}(\zeta_p)$46

These are partial Vandiver theorems because they enforce Vandiver-like local $K=\mathbb{Q}(\zeta_p)$47-th power conditions on specific families of primes associated with a hypothetical FLT2 counterexample, rather than on the full class group (Queme, 2011).

5. Gauss-sum criteria, governing fields, and function-field analogues

The Gauss-sum paper gives a different, and explicitly named, partial Vandiver framework. For a prime $K=\mathbb{Q}(\zeta_p)$48, let $K=\mathbb{Q}(\zeta_p)$49 be of order $K=\mathbb{Q}(\zeta_p)$50, let $K=\mathbb{Q}(\zeta_p)$51 be the classical Gauss sum, and let

$K=\mathbb{Q}(\zeta_p)$52

For each even $K=\mathbb{Q}(\zeta_p)$53, the paper defines the set of exponents of $K=\mathbb{Q}(\zeta_p)$54-primarity

$K=\mathbb{Q}(\zeta_p)$55

and the set of exponents of $K=\mathbb{Q}(\zeta_p)$56-irregularity

$K=\mathbb{Q}(\zeta_p)$57

Theorem 1.2 then states that Vandiver’s conjecture holds if and only if there exists $K=\mathbb{Q}(\zeta_p)$58 such that

$K=\mathbb{Q}(\zeta_p)$59

and equivalently if and only if there exist finitely many primes $K=\mathbb{Q}(\zeta_p)$60 such that

$K=\mathbb{Q}(\zeta_p)$61

The paper explicitly presents these criteria as “partial Vandiver theorems” in the sense that they give strong, efficiently checkable conditions and heuristics making counterexamples extremely constrained (Gras, 2018).

A related extension appears in the work of Gras and Quême on governing fields $K=\mathbb{Q}(\zeta_p)$62. There the key Vandiver-style hypothesis is that for a prime ideal $K=\mathbb{Q}(\zeta_p)$63 in $K=\mathbb{Q}(\zeta_p)$64,

$K=\mathbb{Q}(\zeta_p)$65

If there exists at least one prime $K=\mathbb{Q}(\zeta_p)$66 with $K=\mathbb{Q}(\zeta_p)$67, $K=\mathbb{Q}(\zeta_p)$68, and this condition holds, then the first case of FLT, or of SFLT under the supplementary condition $K=\mathbb{Q}(\zeta_p)$69, holds for $K=\mathbb{Q}(\zeta_p)$70. If there exist infinitely many such primes $K=\mathbb{Q}(\zeta_p)$71, the second case holds as well. This is a partial Vandiver theorem in a different cyclotomic field: it replaces full information on $K=\mathbb{Q}(\zeta_p)$72 by a specific plus/minus class-field condition in $K=\mathbb{Q}(\zeta_p)$73 (Gras et al., 2011).

Over function fields, the picture changes sharply. Taelman’s Herbrand–Ribet theorem controls the range $K=\mathbb{Q}(\zeta_p)$74, but the naively extended Kummer–Vandiver-type statement is false. The paper constructs explicit counterexamples with

$K=\mathbb{Q}(\zeta_p)$75

for exponents $K=\mathbb{Q}(\zeta_p)$76 not divisible by $K=\mathbb{Q}(\zeta_p)$77. The consequence is that in the function-field setting only genuinely partial Vandiver-type results remain valid, and the full analogue fails (Anglès et al., 2011).

6. Status, interpretation, and limits of the terminology

In this body of work, “partial Vandiver theorem” is therefore best understood as a structural label for results with one of the following profiles: they isolate a character component of a class group, they control only primes arising from a putative FLT2 or $K=\mathbb{Q}(\zeta_p)$78 counterexample, they prescribe splitting in a specific Kummer extension generated by cyclotomic units, or they provide criteria equivalent to Vandiver for a fixed prime $K=\mathbb{Q}(\zeta_p)$79 without proving the conjecture uniformly in $K=\mathbb{Q}(\zeta_p)$80.

The label also covers intermediate structural equalities. In the K-theoretic approach, the paper defines invariants $K=\mathbb{Q}(\zeta_p)$81 and proves relations such as

$K=\mathbb{Q}(\zeta_p)$82

before concluding with the theorem $K=\mathbb{Q}(\zeta_p)$83, hence $K=\mathbb{Q}(\zeta_p)$84, which the author interprets as a proof of Vandiver’s conjecture. The same source explicitly notes, however, that in the broader number theory community a complete proof of Vandiver’s conjecture is not accepted as known and that the conjecture remains open. Accordingly, the equalities and local-global characterizations preceding the final claim are the parts most naturally read as partial Vandiver results (Stolin, 2020).

A common misconception is to treat every Vandiver-type statement as if it asserted the full conjecture $K=\mathbb{Q}(\zeta_p)$85. The cited literature shows a more differentiated landscape. Some results are purely conditional on $K=\mathbb{Q}(\zeta_p)$86-principality. Some are equivalent reformulations for a fixed $K=\mathbb{Q}(\zeta_p)$87. Some are heuristic or probabilistic. Some, especially in the function-field setting, show that only partial analogues can hold. The unifying feature is not the proof of Vandiver’s conjecture itself, but the extraction of strong arithmetic consequences from restricted class-group or residue-symbol input.

Taken together, these papers place the partial Vandiver theorem at the interface of cyclotomic units, Hilbert $K=\mathbb{Q}(\zeta_p)$88-class field theory, Kummer extensions, Gauss sums, and FLT-type Diophantine constraints. The resulting statements are narrower than the full conjecture, but they are often more explicit: they identify concrete congruences, concrete splitting criteria, and concrete families of primes whose behavior would be forced by any hypothetical counterexample to $K=\mathbb{Q}(\zeta_p)$89, FLT2, or Vandiver’s conjecture itself (Quême, 2013, Queme, 2011, Gras, 2018, Gras et al., 2011).