A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

116 CHAPTER 15. VALUATIONS
of a valuation. Here’s why: Recall that Axiom 3 for a non-archimedean valuation
on K asserts that whenever a ∈ K and |a| ≤ 1, then |a + 1| ≤ 1. Set a = p − 1,
where p = p(t) ∈ K[t] is an irreducible polynomial. Then |a| = c 0 = 1, since
ordp(p − 1) = 0. However, |a + 1| = |p − 1 + 1| = |p| = c 1 < 1, since ordp(p) = 1. If
we take c > 1 instead of c < 1, as I propose, then |p| = c 1 > 1, as required.
Note the (albeit imperfect) analogy between K = k(t) and Q. If s = t −1 , so
k(t) = k(s), the valuation | · | ∞ is of the type (15.3.3) belonging to the irreducible
polynomial p(s) = s.
The reader is urged to prove the following lemma as a homework problem.
Lemma 15.3.5. The only nontrivial valuations on k(t) which are trivial on k are
equivalent to the valuation (15.3.3) or (15.3.4).
For example, if k is a finite field, there are no nontrivial valuations on k, so the
only nontrivial valuations on k(t) are equivalent to (15.3.3) or (15.3.4).