{"id":10994,"date":"2023-12-03T00:17:21","date_gmt":"2023-12-03T00:17:21","guid":{"rendered":"https:\/\/science-hub.click\/p%E9%80%B2%E6%95%B0-%E5%AE%9A%E7%BE%A9\/"},"modified":"2023-12-03T00:17:21","modified_gmt":"2023-12-03T00:17:21","slug":"p%E9%80%B2%E6%95%B0-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=10994","title":{"rendered":"p \u9032\u6570 – \u5b9a\u7fa9"},"content":{"rendered":"

\u6570\u5b66\u3067\u306f\u3001 p<\/i>\u9032\u6570\u306f<\/strong>\u4f53\u306e\u8981\u7d20\u3067\u3059\u3002

$$ {\\mathbb Q_p} $$<\/div> p<\/i> – \u9032\u6570\u3002\u3053\u3053\u3067\u3001 p<\/i>\u306f\u6307\u5b9a\u3055\u308c\u305f\u7d20\u6570\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001 2\u9032<\/i>\u6570\u3001 3\u9032<\/i>\u6570\u306a\u3069\u306b\u3064\u3044\u3066\u8a71\u3057\u307e\u3059\u3002<\/p>

\u907a\u4f53

$$ {\\mathbb Q_p} $$<\/div> p<\/i>\u9032\u6570\u306f\u30d5\u30a3\u30fc\u30eb\u30c9\u88dc\u5b8c\u306b\u3088\u3063\u3066\u69cb\u7bc9\u3055\u308c\u307e\u3059
$$ {\\mathbb Q} $$<\/div>\u6709\u7406\u6570\u306bp<\/i> -adic \u30ce\u30eb\u30e0<\/strong>\u3068\u547c\u3070\u308c\u308b\u7279\u5b9a\u306e\u30ce\u30eb\u30e0\u304c\u88c5\u5099\u3055\u308c\u3066\u304a\u308a\u3001 |<\/span>\u3067\u793a\u3055\u308c\u307e\u3059\u3002 \u3002 | p<\/i><\/sub><\/span> \u3002\u3042\u308b\u610f\u5473<\/a><\/span>\u3001\u8eab\u4f53\u306f
$$ {\\mathbb Q_p} $$<\/div>\u4f53\u306b\u95a2\u4fc2\u3057\u3066\u3044\u308b
$$ {\\R} $$<\/div>\u5b9f\u6570\u3002\u8003\u616e\u3055\u308c\u308b\u30ce\u30eb\u30e0\u304c\u901a\u5e38\u306e\u7d76\u5bfe\u5024\u3067\u3042\u308b\u5834\u5408\u3001\u3053\u308c\u306f\u6709\u7406\u6570\u4f53\u306e\u5b8c\u6210\u3067\u3082\u3042\u308a\u307e\u3059\u3002<\/p>

p<\/i>\u9032\u6570\u30d5\u30a3\u30fc\u30eb\u30c9\u3092\u751f\u307f\u51fa\u3057\u305f\u4e3b\u306a\u52d5\u6a5f\u306f<\/span>\u3001\u6574\u6570\u5217\u30c6\u30af\u30cb\u30c3\u30af\u3092\u6570\u8ad6<\/a><\/span>\u3067\u4f7f\u7528\u3067\u304d\u308b\u3088\u3046\u306b\u3059\u308b\u3053\u3068\u3067\u3057\u305f\u304c\u3001\u305d\u306e\u6709\u7528\u6027\u306f\u73fe\u5728\u3001\u3053\u306e\u67a0\u7d44\u307f\u3092\u306f\u308b\u304b\u306b\u8d85\u3048\u3066\u3044\u307e\u3059\u3002\u3055\u3089\u306b\u30dc\u30c7\u30a3\u306b\u3082\u88c5\u5099\u53ef\u80fd

$$ {\\mathbb Q_p} $$<\/div>\u975e\u30a2\u30eb\u30ad\u30e1\u30c7\u30b9\u306e\u898f\u7bc4\u3002\u305d\u306e\u5f8c\u3001\u901a\u5e38\u306e\u5206\u6790\u3068\u306f\u7570\u306a\u308b\u5206\u6790\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u3092 p \u9032\u5206\u6790\u3068\u547c\u3073\u307e\u3059\u3002<\/p>

\u5de5\u4e8b<\/span><\/h2>

\u5206\u6790\u7684\u30a2\u30d7\u30ed\u30fc\u30c1<\/span><\/h3>

\u5b9f\u6570\u306f\u3001\u6709\u7406\u6570\u306e\u30b3\u30fc\u30b7\u30fc\u6570\u5217\u306e\u540c\u5024\u985e\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u3053\u306e\u5b9a\u7fa9\u306f<\/a><\/span>\u9078\u629e\u3057\u305f\u30e1\u30c8\u30ea\u30c3\u30af\u306b\u4f9d\u5b58\u3059\u308b\u305f\u3081\u3001\u5225\u306e\u30e1\u30c8\u30ea\u30c3\u30af\u3092\u9078\u629e\u3059\u308b\u3068\u3001\u5b9f\u6570\u4ee5\u5916\u306e\u6570\u5024\u304c\u4f5c\u6210\u3055\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u3002\u5b9f\u6570\u306b\u4f7f\u7528\u3055\u308c\u308b\u8a08\u91cf\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u8a08\u91cf\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>

\u4e0e\u3048\u3089\u308c\u305f\u7d20\u6570<\/a><\/span>p<\/i><\/span>\u306b\u5bfe\u3057\u3066\u3001\u6b21\u306ep<\/i>\u9032\u30ce\u30eb\u30e0<\/strong>\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002

$$ {\\mathbb Q} $$<\/div>\u6b21\u306e\u3088\u3046\u306b\uff1a<\/p>
\u975e\u30bc\u30ed\u6574\u6570a<\/i>\u306ep<\/i>\u9032\u8a55\u4fa1\u306f<\/a><\/span>\u3001 a<\/i>\u3092\u7d20\u56e0\u6570\u306e\u7a4d\u306b\u5206\u89e3\u3059\u308b<\/a><\/span>\u969b\u306ep<\/i>\u306e\u6307\u6570<\/a><\/span>\u3067\u3042\u308b\u3053\u3068\u3092\u601d\u3044\u51fa\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/dd>
\u6b21\u306b\u3001\u6b21\u306e\u3088\u3046\u306b\u5c0b\u306d\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u3001\u30bc\u30ed\u4ee5\u5916\u306e\u6709\u7406\u6570<\/a><\/span>\u306e<\/a><\/span>\u8a55\u4fa1\u3092\u69cb\u7bc9\u3067\u304d\u307e\u3059\u3002
$$ {v_p\\left(\\frac ab \\right) = v_p(a) – v_p(b)} $$<\/div> \u3002<\/dd><\/dl><\/dd>
\u3053\u306e\u5b9a\u7fa9\u304c\u5408\u7406\u7684\u306b\u9078\u629e\u3055\u308c\u305f\u3082\u306e\u306e\u4ee3\u8868\u304b\u3089\u72ec\u7acb\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u7c21\u5358\u306b\u8a3c\u660e\u3057\u307e\u3059\u3002<\/dd>
p<\/i>\u9032\u30ce\u30eb\u30e0| r<\/i> |<\/span>\u975e\u30bc\u30ed\u306e\u6709\u7406\u6570r<\/i><\/span>\u306ep<\/i><\/sub><\/span>\u306f\u4fa1\u5024\u304c\u3042\u308b
$$ {p^{-v_p(r)}} $$<\/div> \u3002<\/dd>
r<\/i>\u304c\u30bc\u30ed\u306e\u5834\u5408\u3001 |<\/span>\u3092\u8a2d\u5b9a\u3057\u307e\u3059\u3002 r<\/i> | p<\/i><\/sub> = 0<\/span> \u3002\u3053\u306e\u62e1\u5f35\u306f\u3001k \u306e\u4efb\u610f\u306e\u5024\u306b\u5bfe\u3057\u3066 0 \u304cp<\/i> k<\/i><\/sup><\/span>\u3067\u5272\u308a\u5207\u308c\u308b\u305f\u3081\u30010 \u306e\u8a55\u4fa1\u306f\u7121\u9650\u306b\u306a\u308b\u3068\u3044\u3046\u8003\u3048\u3068\u4e92\u63db\u6027\u304c\u3042\u308a\u307e\u3059\u3002<\/dd><\/dl>

\u3042\u308b\u610f\u5473\u3067\u306f\u3001 r \u304c<\/i><\/span>p<\/i><\/span>\u3067\u5272\u308a\u5207\u308c\u308b\u307b\u3069\u3001\u305d\u306ep<\/i>\u9032\u30ce\u30eb\u30e0\u306f\u5c0f\u3055\u304f\u306a\u308a\u307e\u3059 (\u3053\u308c\u306f\u96e2\u6563\u8a55\u4fa1\u306e\u7279\u6b8a\u306a\u30b1\u30fc\u30b9\u3067\u3042\u308a\u3001\u4ee3\u6570\u30c4\u30fc\u30eb\u3067\u3059<\/a><\/span>!)\u3002<\/p>

\u305f\u3068\u3048\u3070\u3001

$$ {r = {63 \\over 550} = 2^{-1}\\times 3^2\\times 5^{-2}\\times 7\\times 11^{-1}} $$<\/div> : <\/p>
$$ {|r|_2=2\\,} $$<\/div><\/dd>
$$ {|r|_3={1 \\over 9}\\,} $$<\/div><\/dd>
$$ {|r|_5=25\\,} $$<\/div><\/dd>
$$ {|r|_7={1\\over 7}\\,} $$<\/div><\/dd>
$$ {|r|_{11}=11\\,} $$<\/div><\/dd>
$$ {|r|_p=1\\,} $$<\/div>\u4ed6\u306e\u7d20\u6570\u306e\u5834\u5408\u3002<\/dd><\/dl>

\u3053\u306e\u30a2\u30d7\u30ea\u30b1\u30fc\u30b7\u30e7\u30f3\u304c\u6a19\u6e96\u306e\u3059\u3079\u3066\u306e\u7279\u6027\u3092\u5099\u3048\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002\u3042\u3089\u3086\u308b\uff08\u81ea\u660e\u3067\u306f\u306a\u3044\uff09\u6a19\u6e96\u304c

$$ {\\mathbb Q} $$<\/div>\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9 \u30ce\u30eb\u30e0\u307e\u305f\u306fp<\/i> -\u9032\u30ce\u30eb\u30e0 (\u30aa\u30b9\u30c8\u30ed\u30d5\u30b9\u30ad\u30fc\u306e\u5b9a\u7406) \u3068\u540c\u7b49\u3067\u3059\u3002 p<\/i>\u9032\u30ce\u30eb\u30e0\u306f\u8a08\u91cfdp<\/i>\u3092<\/i><\/sub><\/span>\u5b9a\u7fa9\u3057\u307e\u3059\u3002
$$ {\\mathbb Q} $$<\/div>\u5c0b\u306d\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff1a<\/p>
dp<\/i> ( x<\/i> , y<\/i><\/sub> ) = |<\/i> x<\/i> \u2212 y<\/i> | p<\/i><\/sub><\/span><\/dd><\/dl>

\u4f53

$$ {\\mathbb Q_p} $$<\/div> p<\/i>\u9032\u6570\u306f\u8a08\u91cf\u7a7a\u9593<\/a><\/span>\u306e\u5b8c\u6210\u3068\u3057\u3066\u5b9a\u7fa9\u3067\u304d\u307e\u3059 (
$$ {\\mathbb Q} $$<\/div> \u3001 d<\/i> p<\/i><\/sub><\/span> )\u3002\u305d\u306e\u8981\u7d20\u306f\u30b3\u30fc\u30b7\u30fc\u6570\u5217\u306e\u540c\u5024\u30af\u30e9\u30b9\u3067\u3042\u308a\u30012 \u3064\u306e\u6570\u5217\u306e\u5dee\u304c0<\/a><\/span>\u306b\u53ce\u675f\u3059\u308b\u5834\u5408\u30012 \u3064\u306e\u6570\u5217\u306f\u540c\u7b49\u3067\u3042\u308b\u3068\u8a00\u308f\u308c\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u3001\u30d5\u30a3\u30fc\u30eb\u30c9\u3067\u3082\u3042\u308a\u3001\u4ee5\u4e0b\u3092\u542b\u3080\u5b8c\u5168\u306a\u8a08\u91cf\u7a7a\u9593\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002
$$ {\\mathbb Q} $$<\/div> \u3002<\/p>

\u3053\u306e\u69cb\u9020\u306b\u3088\u308a\u3001\u305d\u306e\u7406\u7531\u304c\u7406\u89e3\u3067\u304d\u307e\u3059\u3002

$$ {\\mathbb Q_p} $$<\/div>\u306f\u7b97\u8853\u7684\u306b\u985e\u4f3c\u3057\u305f\u3082\u306e\u3067\u3059
$$ {\\mathbb R} $$<\/div> \u3002<\/p>
\n\"<\/figure>

\u4ee3\u6570\u7684\u30a2\u30d7\u30ed\u30fc\u30c1<\/span><\/h3>

\u3053\u306e\u4ee3\u6570\u7684\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u306f\u3001\u307e\u305ap<\/i>\u9032\u6574\u6570\u306e\u30ea\u30f3\u30b0\u3092\u5b9a\u7fa9\u3057\u3001\u6b21\u306b\u3053\u306e\u30ea\u30f3\u30b0\u306e\u5206\u6570\u30d5\u30a3\u30fc\u30eb\u30c9\u3092\u69cb\u7bc9\u3057\u3066p<\/i>\u9032\u6570\u306e\u30d5\u30a3\u30fc\u30eb\u30c9\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002<\/p>

p<\/i>\u9032\u6574\u6570\u306e\u30ea\u30f3\u30b0\u3092\u5b9a\u7fa9\u3057\u307e\u3059

$$ {\\mathbb Z_p} $$<\/div>\u30ea\u30f3\u30b0\u306e\u5c04\u5f71\u9650\u754c\u306e\u3088\u3046\u306b
$$ {\\mathbb Z\/p^n\\mathbb Z} $$<\/div> \u3002 p<\/i>\u9032\u6574\u6570\u306f\u30b7\u30fc\u30b1\u30f3\u30b9\u306b\u306a\u308a\u307e\u3059
$$ {(a_n)_{n\\ge 1}} $$<\/div>\u306e\u3088\u3046\u306a
$$ {a_n \\in \\mathbb Z\/p^n\\mathbb Z} $$<\/div>\u305d\u3057\u3066\u3001 n<\/i> < m<\/i>\u306e<\/span>\u5834\u5408\u3001 a<\/i> n<\/i><\/sub> = a<\/i> m<\/i><\/sub> [ p<\/i> n<\/i><\/sup> ]<\/span>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>

\u305f\u3068\u3048\u3070\u30012 \u9032\u6570\u306e35 \u306f<\/span>\u6b21\u306e\u30b7\u30fc\u30b1\u30f3\u30b9\u306b\u306a\u308a\u307e\u3059\u3002

$$ {(1, 3, 3, 3, 3, 35, 35, 35 \\ldots)} $$<\/div> \u3002<\/p>

\u3053\u306e\u3088\u3046\u306a\u30b7\u30fc\u30b1\u30f3\u30b9\u306e\u52a0\u7b97<\/a><\/span>\u3068\u4e57\u7b97\u306f<\/a><\/span>\u3001\u30e2\u30b8\u30e5\u30ed<\/a><\/span>\u6f14\u7b97\u5b50<\/a><\/span>\u3092\u4f7f\u7528\u3057\u3066\u4ea4\u63db\u3067\u304d\u308b\u305f\u3081\u3001\u660e\u78ba\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059 (\u30e2\u30b8\u30e5\u30e9\u30fc\u7b97\u8853\u3092\u53c2\u7167)\u3002\u3055\u3089\u306b\u3001\u6700\u521d\u306e\u8981\u7d20\u304c 0 \u3067\u306a\u3044\u30b7\u30fc\u30b1\u30f3\u30b9( a<\/i> n<\/i><\/sub> )<\/span>\u306f\u3001\u9006\u884c\u5217<\/a><\/span>\u3092\u6301\u3061\u307e\u3059\u3002<\/p>

p<\/i>\u9032\u6574\u6570\u306e\u30ea\u30f3\u30b0\u306b\u306f\u30bc\u30ed\u306e\u7d04\u6570\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u306e\u5206\u6570\u4f53\u3092\u8003\u616e\u3057\u3066\u6b21\u306e\u4f53\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002

$$ {\\mathbb Q_p} $$<\/div> p<\/i>\u9032\u6570\u3002<\/p>
\n\"<\/figure>

\u30d8\u30f3\u30bc\u30eb\u306e\u6b63\u6e96\u5206\u89e3<\/span><\/h2>

p\u3092<\/i><\/span>\u7d20\u6570\u3068\u3059\u308b\u3002\u306e\u30bc\u30ed\u4ee5\u5916\u306e\u8981\u7d20r<\/i><\/span>

$$ {\\mathbb Q_p} $$<\/div> (\u7279\u306b\u3001
$$ {\\mathbb Q} $$<\/div> ) \u306f\u6b21\u306e\u5f62\u5f0f\u3067\u72ec\u81ea\u306b\u8a18\u8ff0\u3055\u308c\u307e\u3059\u3002 <\/p>
$$ {r = \\sum_{i=k}^\\infty a_i p^i} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u307e\u305f\u306f

$$ {k \\in \\Z} $$<\/div> a<\/i> i \u306f<\/i><\/sub><\/span>0<\/span>\u304b\u3089p<\/i> \u2212 1<\/span>\u307e\u3067\u306e\u6574\u6570\u3067\u3059\u3002\u3053\u306e\u8a18\u8ff0\u306f\u3001 r<\/i><\/span>\u3092p<\/i>\u9032\u6570\u3068\u3057\u3066\u6b63\u898f\u306b\u5206\u89e3\u3057\u305f\u3082\u306e\u3067\u3059\u3002<\/p>

\u3053\u306e\u7cfb\u5217\u306f\u3001 p<\/i>\u9032\u30e1\u30c8\u30ea\u30c3\u30af\u306b\u5f93\u3063\u3066\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>

\u6ce8\u610f\u3057\u307e\u3059

$$ {\\Z_p} $$<\/div>\u306e\u3059\u3079\u3066\u306e\u8981\u7d20
$$ {\\mathbb Q_p} $$<\/div>\u306e\u3088\u3046\u306a
$$ {k\\ge 0} $$<\/div>\u3053\u308c\u3089\u3092\u5408\u308f\u305b\u3066p<\/i>\u9032\u6574\u6570\u3068\u547c\u3073\u307e\u3059\u3002
$$ {\\Z_p} $$<\/div>\u306e\u90e8\u5206\u74b0\u3067\u3059
$$ {\\mathbb Q_p} $$<\/div> \u3002 p<\/i>\u9032\u6574\u6570\u306f\u3001\u57fa\u6570p<\/i>\u306e\u6570\u5b57\u306e\u5de6\u5074\u306b\u3042\u308b\u7121\u9650\u30b7\u30fc\u30b1\u30f3\u30b9\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001\u4ed6\u306e\u8981\u7d20\u306f
$$ {\\mathbb Q_p} $$<\/div> \u3001\u5c0f\u6570\u70b9\u306e\u53f3\u5074\u306b\u6709\u9650\u306e\u6841\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u3053\u306e\u8a18\u8ff0\u306f\u3001\u5b9f\u6570\u306e\u8a18\u8ff0\u3067\u901a\u5e38\u906d\u9047\u3059\u308b\u3082\u306e\u3068\u306f\u9006\u306e\u65b9\u6cd5\u3067\u6a5f\u80fd\u3057\u307e\u3059\u3002<\/p>

\u305f\u3068\u3048\u3070\u3001 p<\/i> = 2<\/span>\u306e\u5834\u5408: <\/p>