**Hien Ha Thursday ****May 12, 3pm Keller ****302**

Tile: p-adic numbers

Abstract: The field of real numbers is completed from the field of rational numbers with respect to the distance metric. The heart of the completion process is the limit of Cauchy sequences. Recall that a metric space (X, d) is complete if every Cauchy sequence in X converges to a point in X. We know that Q is not complete with respect to the distance metric. For example, the Cauchy sequence of rational numbers 3 , 31/10 , 314 /100 , 3141/ 1000 …. is converging to π which is not in Q . Filling all convergent points of Cauchy sequences we get R. The field of p-adic numbers Qp is also completed from Q with respect to a different metric called the p-adic metric which is induced from the p-adic norm. We also make use of the convergence of the Cauchy sequences for this process. In this project we will see how Qp is completed from Q with respect to the p-adic norm. We also describe two trees associated with Qp: the trees of balls in the field Qp and the trees of lattices in the vector space Qp × Qp.

# Calendar

May

9

Mon

May 9 @ 2:30 pm – 3:30 pm

May

10

Tue

May 10 @ 2:30 pm – 3:30 pm

May

12

Thu

May 12 @ 3:00 pm – 4:00 pm

May

25

Wed

May 25 @ 3:00 pm – 4:00 pm

Jul

12

Tue

Jul 12 @ 11:00 am – 12:00 pm

Aug

5

Fri

Aug 5 @ 2:00 pm – 3:00 pm

Aug

18

Thu

Aug 18 @ 3:30 pm – 4:30 pm

Aug

22

Mon

Aug 22 all-day