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WLPowers of a unit class come from integer powers

Lemma·L79

A power of a unit class is the class of the corresponding integer power.

Let with and let be a unit. For every , where the left is the group power in the group of units and is the integer power.
In words
The k-th power of a unit class equals the class of the k-th power of its representative.
Rests onF02F03F04F05F06F08F09F11F12F13F14A01A02A06A08
Never needed: F10 · A03 · A04 · A05 · A07 · A09 (computed from the citation graph, not asserted).

Proof

  1. 1
    By induction on (T05). Write ; the group operation is class multiplication and its identity is (T61).
  2. 2
    Base . (D043, identity). And (D119), so . The two agree.
  3. 3
    Step. Assume . Then using D043 (so ), the hypothesis, D116, and D119 (so ). By T05 the identity holds for all .
∎

Remarks

The group power (repeated class multiplication, D043) and the integer power carried into (D119) satisfy the same recursion, so they coincide on units. This is the bridge that converts the group-theoretic identity from L36 into the integer congruence , that is Euler's theorem, and likewise powers Fermat's little theorem.
Wikipedia ↗map_pow ↗

Used by

T62
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