Integer Less One divides Power Less One/Corollary
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Corollary to Integer Less One divides Power Less One
Let $m, n, q \in \Z_{>0}$.
Let:
- $m \divides n$
where $\divides$ denotes divisibility.
Then:
- $\paren {q^m - 1} \divides \paren {q^n - 1}$
Converse to Corollary
Let $m, n, q \in \Z_{>0}$.
Let
- $\paren {q^m - 1} \divides \paren {q^n - 1}$
where $\divides$ denotes divisibility.
Then:
- $m \divides n$
Proof
- $m \divides n$
By definition of divisibility:
- $\exists k \in \Z: k m = n$
Thus:
- $q^n = q^{k m} = \paren {q^m}^k$
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Then by Integer Less One divides Power Less One:
- $\paren {q^m - 1} \divides \paren {\paren {q^m}^k - 1}$
Hence the result.
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $5$