адический порядок

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${\ displaystyle r = p_ {1} ^ {\ nu _ {p_ {1}}} p_ {2} ^ {\ nu _ {p_ {2}}} \ cdots p_ {k} ^ {\ nu _ {p_ {k}}} = \ prod _ {i = 1} ^ {k} p_ {i} ^ {\ nu _ {p_ {i}}},}$

#1075;#1076;#1077; #1087;#1088;#1086;#1089;#1090;#1099;#1077; #1095;#1080;#1089;#1083;#1072; #1080; #1103;#1074;#1083;#1103;#1102;#1090;#1089;#1103; (#1091;#1085;#1080;#1082;#1072;#1083;#1100;#1085;#1099;#1077;) #1094;#1077;#1083;#1099;#1077; #1095;#1080;#1089;#1083;#1072; (#1089;#1095;#1080;#1090;#1072;#1077;#1090;#1089;#1103; 0 #1076;#1083;#1103; #1074;#1089;#1077;#1093; #1087;#1088;#1086;#1089;#1090;#1099;#1093; #1095;#1080;#1089;#1077;#1083;, #1085;#1077; #1074;#1093;#1086;#1076;#1103;#1097;#1080;#1077; #1074; #1075; #1090;#1072;#1082;, #1095;#1090;#1086;). ${\ displaystyle p_ {1} lt;p_ {2} lt;\ dotsb lt;p_ {k}}$${\ displaystyle \ nu _ {p_ {i}}}$ ${\ displaystyle \ nu _ {p_ {i}} (r) = \ nu _ {p_ {i}} (a) - \ nu _ {p_ {i}} (b)}$

#1069;#1090;#1086;#1090; p -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1080;#1081; #1087;#1086;#1088;#1103;#1076;#1086;#1082; #1089;#1086;#1089;#1090;#1072;#1074;#1083;#1103;#1077;#1090; (#1072;#1076;#1076;#1080;#1090;#1080;#1074;#1085;#1086; #1079;#1072;#1087;#1080;#1089;#1072;#1085;#1085;#1091;#1102;) #1086;#1094;#1077;#1085;#1082;#1091;, #1090;#1072;#1082; #1085;#1072;#1079;#1099;#1074;#1072;#1077;#1084;#1091;#1102; p -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1091;#1102; #1086;#1094;#1077;#1085;#1082;#1091;, #1082;#1086;#1090;#1086;#1088;#1072;#1103; #1087;#1088;#1080; #1084;#1091;#1083;#1100;#1090;#1080;#1087;#1083;#1080;#1082;#1072;#1090;#1080;#1074;#1085;#1086;#1081; #1079;#1072;#1087;#1080;#1089;#1080; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1072;#1085;#1072;#1083;#1086;#1075;#1086;#1084; #1093;#1086;#1088;#1086;#1096;#1086; #1080;#1079;#1074;#1077;#1089;#1090;#1085;#1086;#1075;#1086; #1086;#1073;#1099;#1095;#1085;#1086;#1075;#1086; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1086;#1075;#1086; #1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103;. #1054;#1073;#1072; #1090;#1080;#1087;#1072; #1086;#1094;#1077;#1085;#1086;#1082; #1084;#1086;#1075;#1091;#1090; #1080;#1089;#1087;#1086;#1083;#1100;#1079;#1086;#1074;#1072;#1090;#1100;#1089;#1103; #1076;#1083;#1103; #1079;#1072;#1074;#1077;#1088;#1096;#1077;#1085;#1080;#1103; #1087;#1086;#1083;#1103; #1088;#1072;#1094;#1080;#1086;#1085;#1072;#1083;#1100;#1085;#1099;#1093; #1095;#1080;#1089;#1077;#1083;, #1075;#1076;#1077; #1079;#1072;#1074;#1077;#1088;#1096;#1077;#1085;#1080;#1077; #1089; #1087;#1086;#1084;#1086;#1097;#1100;#1102; p -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1086;#1081; #1086;#1094;#1077;#1085;#1082;#1080; #1087;#1088;#1080;#1074;#1086;#1076;#1080;#1090; #1082; #1087;#1086;#1083;#1102; p -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1080;#1093; #1095;#1080;#1089;#1077;#1083; #8474; pthinsp; (#1086;#1090;#1085;#1086;#1089;#1080;#1090;#1077;#1083;#1100;#1085;#1086; #1074;#1099;#1073;#1088;#1072;#1085;#1085;#1086;#1075;#1086; #1087;#1088;#1086;#1089;#1090;#1086;#1075;#1086; #1095;#1080;#1089;#1083;#1072; p), #1090;#1086;#1075;#1076;#1072; #1082;#1072;#1082; #1079;#1072;#1074;#1077;#1088;#1096;#1077;#1085;#1080;#1077; #1089; #1087;#1086;#1084;#1086;#1097;#1100;#1102; #1086;#1073;#1099;#1095;#1085;#1099;#1077; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1099;#1077; #1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103; #1087;#1088;#1080;#1074;#1086;#1076;#1103;#1090; #1074; #1087;#1086;#1083;#1077; #1076;#1077;#1081;#1089;#1090;#1074;#1080;#1090;#1077;#1083;#1100;#1085;#1099;#1093; #1095;#1080;#1089;#1077;#1083; #8477;.

#1056;#1072;#1089;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1080;#1077; #1085;#1072;#1090;#1091;#1088;#1072;#1083;#1100;#1085;#1099;#1093; #1095;#1080;#1089;#1077;#1083; #1087;#1086; #1080;#1093; 2-#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1086;#1084;#1091; #1087;#1086;#1088;#1103;#1076;#1082;#1091; #1089; #1089;#1086;#1086;#1090;#1074;#1077;#1090;#1089;#1090;#1074;#1091;#1102;#1097;#1080;#1084;#1080; #1089;#1090;#1077;#1087;#1077;#1085;#1103;#1084;#1080; #1076;#1074;#1086;#1081;#1082;#1080; #1074; #1076;#1077;#1089;#1103;#1090;#1080;#1095;#1085;#1086;#1081; #1089;#1080;#1089;#1090;#1077;#1084;#1077;. #1059; #1085;#1091;#1083;#1103; #1074;#1089;#1077;#1075;#1076;#1072; #1073;#1077;#1089;#1082;#1086;#1085;#1077;#1095;#1085;#1099;#1081; #1087;#1086;#1088;#1103;#1076;#1086;#1082;.

• 1 #1054;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1080;#1077; #1080; #1089;#1074;#1086;#1081;#1089;#1090;#1074;#1072;
  • 1.1 #1062;#1077;#1083;#1099;#1077; #1095;#1080;#1089;#1083;#1072;
  • 1.2 #1056;#1072;#1094;#1080;#1086;#1085;#1072;#1083;#1100;#1085;#1099;#1077; #1095;#1080;#1089;#1083;#1072;
• 2 p -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1072;#1103; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1072;#1103; #1074;#1077;#1083;#1080;#1095;#1080;#1085;#1072;
• 3 #1057;#1084;. #1058;#1072;#1082;#1078;#1077;
• 4 #1089;#1089;#1099;#1083;#1082;#1080;

#1055;#1091;#1089;#1090;#1100; p - #1087;#1088;#1086;#1089;#1090;#1086;#1077; #1095;#1080;#1089;#1083;#1086;.

#1062;#1077;#1083;#1099;#1077; #1095;#1080;#1089;#1083;#1072;

#1056; -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1072;#1103; #1087;#1086;#1088;#1103;#1076;#1086;#1082; #1080;#1083;#1080; #1088; -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1072;#1103; #1086;#1094;#1077;#1085;#1082;#1080; #1076;#1083;#1103; #8484; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1092;#1091;#1085;#1082;#1094;#1080;#1103;

${\ displaystyle \ nu _ {p}: \ mathbb {Z} \ to \ mathbb {N}}$

#1086;#1087;#1088;#1077;#1076;#1077;#1083;#1103;#1077;#1090;#1089;#1103;

${\ displaystyle \ nu _ {p} (n) = {\ begin {case} \ mathrm {max} \ {k \ in \ mathbb {N}: p ^ {k} \ mid n \} amp; {\ text { if}} n \ neq 0 \\\ infty amp; {\ text {if}} n = 0, \ end {case}}}$

#1075;#1076;#1077; #1086;#1073;#1086;#1079;#1085;#1072;#1095;#1072;#1077;#1090; #1085;#1072;#1090;#1091;#1088;#1072;#1083;#1100;#1085;#1099;#1077; #1095;#1080;#1089;#1083;#1072;. ${\ Displaystyle \ mathbb {N}}$

#1053;#1072;#1087;#1088;#1080;#1084;#1077;#1088;, #1080; #1089; #1090;#1077;#1093; #1087;#1086;#1088;. ${\ displaystyle \ nu _ {3} (- 45) = 2}$${\ displaystyle \ nu _ {5} (- 45) = 1}$${\ displaystyle | {-45} | = 45 = 3 ^ {2} \ cdot 5 ^ {1}}$

#1054;#1073;#1086;#1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103; #1080;#1085;#1086;#1075;#1076;#1072; #1080;#1089;#1087;#1086;#1083;#1100;#1079;#1091;#1102;#1090;#1089;#1103; #1076;#1083;#1103; #1086;#1073;#1086;#1079;#1085;#1072;#1095;#1077;#1085;#1080;#1103;. ${\ displaystyle p ^ {k} \ parallel n}$${\ Displaystyle #1082; = \ #1085;#1102; _ {#1088;} (#1087;)}$

#1056;#1072;#1094;#1080;#1086;#1085;#1072;#1083;#1100;#1085;#1086;#1077; #1095;#1080;#1089;#1083;#1086;

#1056; -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1080;#1081; #1079;#1072;#1082;#1072;#1079; #1084;#1086;#1078;#1077;#1090; #1073;#1099;#1090;#1100; #1087;#1088;#1086;#1076;#1083;#1077;#1085; #1074; #1088;#1072;#1094;#1080;#1086;#1085;#1072;#1083;#1100;#1085;#1099;#1077; #1095;#1080;#1089;#1083;#1072; #1082;#1072;#1082; #1092;#1091;#1085;#1082;#1094;#1080;#1080;

${\ displaystyle \ nu _ {p}: \ mathbb {Q} \ to \ mathbb {Z}}$

#1086;#1087;#1088;#1077;#1076;#1077;#1083;#1103;#1077;#1090;#1089;#1103;

${\ displaystyle \ nu _ {p} \ left ({\ frac {a} {b}} \ right) = \ nu _ {p} (a) - \ nu _ {p} (b).}$

#1053;#1072;#1087;#1088;#1080;#1084;#1077;#1088;, #1080; #1089; #1090;#1077;#1093; #1087;#1086;#1088;. ${\ displaystyle \ nu _ {2} {\ bigl (} {\ tfrac {9} {8}} {\ bigr)} = - 3}$${\ displaystyle \ nu _ {3} {\ bigl (} {\ tfrac {9} {8}} {\ bigr)} = 2}$${\ displaystyle {\ tfrac {9} {8}} = {\ tfrac {3 ^ {2}} {2 ^ {3}}}}$

#1053;#1077;#1082;#1086;#1090;#1086;#1088;#1099;#1077; #1089;#1074;#1086;#1081;#1089;#1090;#1074;#1072;:

${\ displaystyle {\ begin {align} \ nu _ {p} (m \ cdot n) amp; = \ nu _ {p} (m) + \ nu _ {p} (n) \\ [5px] \ nu _ {p} (m + n) amp; \ geq \ min {\ bigl \ {} \ nu _ {p} (m), \ nu _ {p} (n) {\ bigr \}}. \ end {#1074;#1099;#1088;#1086;#1074;#1085;#1077;#1085;#1086;} }}$

#1041;#1086;#1083;#1077;#1077; #1090;#1086;#1075;#1086;, #1077;#1089;#1083;#1080;, #1090;#1086; ${\ Displaystyle \ #1085;#1102; _ {#1087;} (#1084;) \ #1085;#1077;#1082; \ #1085;#1102; _ {#1088;} (#1087;)}$

${\ displaystyle \ nu _ {p} (#1084; + n) = \ min {\ bigl \ {} \ nu _ {p} (m), \ nu _ {p} (n) {\ bigr \}}}$

#1075;#1076;#1077; min - #1084;#1080;#1085;#1080;#1084;#1091;#1084; (#1090;.#1077;. #1084;#1077;#1085;#1100;#1096;#1080;#1081; #1080;#1079; #1076;#1074;#1091;#1093;).

#1056; -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1086;#1077; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1086;#1077; #1079;#1085;#1072;#1095;#1077;#1085;#1080;#1077; #1085;#1072; #8474; #1103;#1074;#1083;#1103;#1077;#1090;#1089;#1103; #1092;#1091;#1085;#1082;#1094;#1080;#1077;#1081;

${\ displaystyle | \ cdot | _ {p} \ #1076;#1074;#1086;#1077;#1090;#1086;#1095;#1080;#1077; \ mathbb {Q} \ to \ mathbb {R} _ {\ geq 0}}$

#1086;#1087;#1088;#1077;#1076;#1077;#1083;#1103;#1077;#1090;#1089;#1103;

${\ displaystyle | r | _ {p} = p ^ {- \ nu _ {p} (r)}.}$

#1053;#1072;#1087;#1088;#1080;#1084;#1077;#1088;, #1080;${\ displaystyle | {-45} | _ {3} = {\ tfrac {1} {9}}}$${\ displaystyle {\ bigl |} {\ tfrac {9} {8}} {\ bigr |} _ {2} = 8.}$

#1056; -#1072;#1076;#1080;#1095;#1077;#1089;#1082;#1086;#1075;#1086; #1072;#1073;#1089;#1086;#1083;#1102;#1090;#1085;#1086;#1077; #1079;#1085;#1072;#1095;#1077;#1085;#1080;#1077; #1091;#1076;#1086;#1074;#1083;#1077;#1090;#1074;#1086;#1088;#1103;#1102;#1090; #1089;#1083;#1077;#1076;#1091;#1102;#1097;#1080;#1077; #1089;#1074;#1086;#1081;#1089;#1090;#1074;#1072;.

#1053;#1077;#1086;#1090;#1088;#1080;#1094;#1072;#1090;#1077;#1083;#1100;#1085;#1086;#1089;#1090;#1100;	${\ displaystyle | a | _ {p} \ geq 0}$
#1055;#1086;#1083;#1086;#1078;#1080;#1090;#1077;#1083;#1100;#1085;#1072;#1103; #1086;#1087;#1088;#1077;#1076;#1077;#1083;#1077;#1085;#1085;#1086;#1089;#1090;#1100;	${\ displaystyle | a | _ {p} = 0 \ iff a = 0}$
#1052;#1091;#1083;#1100;#1090;#1080;#1087;#1083;#1080;#1082;#1072;#1090;#1080;#1074;#1085;#1086;#1089;#1090;#1100;	${\ displaystyle | ab | _ {p} = | a | _ {p} | b | _ {p}}$
#1053;#1077;#1072;#1088;#1093;#1080;#1084;#1077;#1076;#1086;#1074;	${\ displaystyle | a + b | _ {p} \ leq \ max \ left (| a | _ {p}, | b | _ {p} \ right)}$

#1057;#1080;#1084;#1084;#1077;#1090;#1088;#1080;#1080; #1089;#1083;#1077;#1076;#1091;#1077;#1090; #1080;#1079; #1084;#1091;#1083;#1100;#1090;#1080;#1087;#1083;#1080;#1082;#1072;#1090;#1080;#1074;#1085;#1086;#1089;#1090;#1080; #1080; #1089;#1091;#1073;#1072;#1076;#1076;#1080;#1090;#1080;#1074;#1085;#1086;#1089;#1090;#1080; #1080;#1079; #1085;#1077;#1072;#1088;#1093;#1080;#1084;#1077;#1076;#1086;#1074;#1086;#1075;#1086; #1085;#1077;#1088;#1072;#1074;#1077;#1085;#1089;#1090;#1074;#1072; #1090;#1088;#1077;#1091;#1075;#1086;#1083;#1100;#1085;#1080;#1082;#1072;. ${\ displaystyle | {-a} | _ {p} = | a | _ {p}}$ ${\ displaystyle | ab | _ {p} = | a | _ {p} | b | _ {p}}$ ${\ displaystyle | a + b | _ {p} \ leq | a | _ {p} + | b | _ {p}}$ ${\ displaystyle | a + b | _ {p} \ leq \ max \ left (| a | _ {p}, | b | _ {p} \ right)}$

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${\ displaystyle \ prod _ {0, p} | x | _ {p} = 1}$

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