(1)MATH-GA Homework 3

MATH-GA 2420.006 : Homework 3; due by Monday February 22 morning (before 10am), late submission is possible, notify by email;

send the solutions to pirutka@cims.nyu.edu 1. Determine a) H(²₃); b) H(¹⁺

√−5

2 ).

2. Let K be a number field and letL/K be a finite extension. Letp⊂ O_K be a prime ideal corresponding to an absolute value v onK. Write

pO_L=Y

w

q^e_w^w

decomposition into (distinct) prime ideals inOL. Let fw be the degree f_w = [O_L/q_w :O_K/p].

(a) Show that #O_L/p= (#O_K/p)^[L:K]

(one can use that O_K =Ze₁ ⊕. . .⊕Ze_n for n = [K :Q] and e_i ∈ O_K, i= 1. . . n, simialrly for L.)

(b) Deduce that [L:K] =P ewfw.

(c) If x∈ O_K show that ord_qwx=e_word_p(x). Deduce that|x|_w =|x|^e_v^wfw. (d) Deduce that for x₀, . . . x_n∈K one has Q

wmax_i|x_i|_w =max_i|x_i|^[L:K]v . (e) Deduce that for P ∈Pⁿ(K)one has H_L(P) = H_K(P)^[L:K].