LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A full-rank unimodular sublattice has index one

Statement

Let $n \geq 1$ , let $L$ be a free abelian group of rank $n$ with an integer-valued symmetric bilinear form whose Gram determinant is $1$ or $- 1$ , and let $f : L \to Z^{n}$ be an isometric homomorphism, where $Z^{n}$ has its standard dot product. Then $f (L) = Z^{n}$ .

Facts & Assumptions

Given: A rank- $n$ free abelian group $L$ , a unimodular integral symmetric form on $L$ , and an isometric homomorphism $f$ to the standard dot-product lattice $Z^{n}$ .

[A1]

The stated Gram determinant has absolute value one; the argument is finite integral linear algebra and uses no choice axiom.

[F1]

If $A \in M_{n} (Z)$ has nonzero determinant, then the subgroup $A Z^{n}$ has index $∣ \det A ∣$ in $Z^{n}$ . The index of a full-rank subgroup of $Z^{n}$ is the absolute determinant of a generating matrix

Proof

1.1givenA1chooseconstruct

Choose an integral basis of $L$ and let $B$ be the integer $n \times n$ matrix whose columns are the images of that basis under $f$ . Since $f$ is isometric, the Gram matrix of $L$ is $B^{T} B$ .

2.1step 1.1algebra

Taking determinants gives $1 = ∣ \det (B^{T} B) ∣ = ∣ \det B ∣^{2}$ , hence $\det B = 1$ or $- 1$ .

3.1F1step 2.1algebra∎

By [F1] and step 2.1, the image subgroup $B Z^{n} = f (L)$ has index $∣ \det B ∣ = 1$ in $Z^{n}$ . The quotient therefore has one element, which means $B Z^{n} = Z^{n}$ and $f (L) = Z^{n}$ .

Depends on

The index of a full-rank subgroup of $\mathbb Z^n$ is the absolute determinant of a generating matrix

Used by

The E8 lattice is even unimodular and is not diagonal Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shalom Eliahou and Eitan Shalom, Number Theory course notes (standard reference, not scraped)