Sign up to save your podcasts
Or
Share Extension of complete discrete valuation fields (results) #ALNT-LB 1.9.R Chapter 1 Section 9 #Algebraic Number Theory # Lecture note Benois
Share to email
Share to Facebook
Share to X
Share to email
Share to Facebook
Share to X
Or
Extension of complete discrete valuation fields (results) #ALNT-LB 1.9.R Chapter 1 Section 9 #Algebraic Number Theory # Lecture note Benois
Let K be a complete discrete valuation field.
Theorem 9.1
Let L/K be a finite separable extension, then
(1) L is (also) a complete discrete valuation field.
(2) O_L coincides with the integral closure of O_K in L.
(3) O_L is a free O_K-module of rank [L:K].
Definition
Let L/K be a finite separable extension of a complete discrete valuation field (hence so is L).
(1) ramification index: pi_K=pi_L^e u; set v_L(pi_L)=1, then e=v_L(pi_K)
(2) residue degree: f=[k_L: k_K].
Theorem 9.2
Let L/K be a finite separable extension of a complete discrete valuation field, then ef=[L:K].
Proposition 9.3
Let L/K be a finite separable extension of a complete discrete valuation field. Let v_K(pi_K)=v_L(pi_L)=1. Then
v_L(x)=1/f v_K(N_{L/K}(x)).
Proposition 9.4
Assume K subset L subset M are complete discrete valuation fields, then
f(M/K)=f(M/L)f(L/K)
e(M/K)=e(M/L)e(L/K).
View all episodes
By LucLet K be a complete discrete valuation field.
Theorem 9.1
Let L/K be a finite separable extension, then
(1) L is (also) a complete discrete valuation field.
(2) O_L coincides with the integral closure of O_K in L.
(3) O_L is a free O_K-module of rank [L:K].
Definition
Let L/K be a finite separable extension of a complete discrete valuation field (hence so is L).
(1) ramification index: pi_K=pi_L^e u; set v_L(pi_L)=1, then e=v_L(pi_K)
(2) residue degree: f=[k_L: k_K].
Theorem 9.2
Let L/K be a finite separable extension of a complete discrete valuation field, then ef=[L:K].
Proposition 9.3
Let L/K be a finite separable extension of a complete discrete valuation field. Let v_K(pi_K)=v_L(pi_L)=1. Then
v_L(x)=1/f v_K(N_{L/K}(x)).
Proposition 9.4
Assume K subset L subset M are complete discrete valuation fields, then
f(M/K)=f(M/L)f(L/K)
e(M/K)=e(M/L)e(L/K).