Proper sublattice

Let $𝐿$ be a rational lattice an suppose $𝑀 = s p a n ℤ {𝛼 𝑖} 𝑛 𝑖 = 1$ is a proper sublattice, and $𝐿 / 𝑀$ has exponent $𝑚$ . Then geo

𝐿 \leq ℤ 1 𝑚 𝑀

and there exists

𝛽 = 𝑛 \sum 𝑖 = 1 𝑚 𝑖 𝑚 𝛼 𝑖 \in 𝐿

for integers $0 \leq 𝑚 𝑖 < 𝑚$ not all equal to zero.¹

Proof

If $𝐿 / 𝑀$ has exponent $𝑚$ , then $𝑚 𝐿 \leq ℤ 𝑀$ , and thus $𝐿 \leq ℤ 1 𝑚 𝑀$ . Moreover
${𝑛 \sum 𝑖 = 1 𝑚 𝑖 𝑚 𝛼 𝑖 : 0 \leq 𝑚 𝑖 < 𝑚}$
forms a set of coset representatives for $(1 𝑚 𝑀) / 𝑀 \geq ℤ 𝐿 / 𝑀$ .

tidy | en | SemBr

2022. Algebraic number theory course notes, ¶2.5, p. 35 ↩

Proper sublattice

Footnotes

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