 Solution

ID: past-exam-of-the-analytic-number-theory-and-elliptic-curves-course-of-the-university-of-oxford/2007/b/5/iii/solution

No prime works. The P-adic valuation induces a well-defined homomorphism

Q_{p}^{*} / (Q_{p}^{*})^{p} ⟶ Z / p Z, [a] ⟼ v_{p} (a) mod p .

The class of

p

maps to one, whereas the class of

p^{p}

maps to zero. They therefore cannot be equal. Equivalently,

p

cannot be a

p

th power because its valuation is not divisible by

p

 New to topics? Read the docs here!