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Share completion and extension of norms (detailed proof) #ALNT-LB 1.4-1.5.DP Chapter 1 Section 4-5 #Algebraic Number Theory # Lecture note Benois
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completion and extension of norms (detailed proof) #ALNT-LB 1.4-1.5.DP Chapter 1 Section 4-5 #Algebraic Number Theory # Lecture note Benois
Definition A normed field (K, ||-||_K) is complete if every Cauchy sequence converges in K.
Theorem 4.1 Let (K, ||-||_K) be a normed field, then there exists a unique field K' (up to isometry) containing K that has a norm ||-||_K' extending ||-||_K and being complete for this norm.
Theorem 5.1 Let (K, ||-||_K) be a complete normed field and L/K an algebraic extension of dimension n, then there exists a unique norm extension over L, given by ||x||_L=||N_{L/K}(x)||_K^{1/n} for any x in L.
Theorem 5.2 Let (K, ||-||_K) be a complete normed field and let V be a finite vector space over K. Then
(1) Any two norms on V are equivalent.
(2) V is complete for any norm.
Corollary1 Let (K, ||-||_K) be a complete normed field and let L/K be a finite Galois extension, then Gal(L/K) acts continuously on L and it preserves norm.
Corollary 2 Let (K, ||-||_K) be a complete normed field and let L/K be a separable extension, then N_{L/K}: L --> K and Tr_{L/K}: L --> K are both continuous.
Remark that N_{L/K}(x)=Prod sigma_i(x) (for all possible sigma_i L/K embeds into K^bar/K) and Tr_{L/K}(x)=Prod sigma_i(x) (for all possible sigma_i L/K embeds into K^bar/K).
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By LucDefinition A normed field (K, ||-||_K) is complete if every Cauchy sequence converges in K.
Theorem 4.1 Let (K, ||-||_K) be a normed field, then there exists a unique field K' (up to isometry) containing K that has a norm ||-||_K' extending ||-||_K and being complete for this norm.
Theorem 5.1 Let (K, ||-||_K) be a complete normed field and L/K an algebraic extension of dimension n, then there exists a unique norm extension over L, given by ||x||_L=||N_{L/K}(x)||_K^{1/n} for any x in L.
Theorem 5.2 Let (K, ||-||_K) be a complete normed field and let V be a finite vector space over K. Then
(1) Any two norms on V are equivalent.
(2) V is complete for any norm.
Corollary1 Let (K, ||-||_K) be a complete normed field and let L/K be a finite Galois extension, then Gal(L/K) acts continuously on L and it preserves norm.
Corollary 2 Let (K, ||-||_K) be a complete normed field and let L/K be a separable extension, then N_{L/K}: L --> K and Tr_{L/K}: L --> K are both continuous.
Remark that N_{L/K}(x)=Prod sigma_i(x) (for all possible sigma_i L/K embeds into K^bar/K) and Tr_{L/K}(x)=Prod sigma_i(x) (for all possible sigma_i L/K embeds into K^bar/K).