2460:
2050:
2771:
2975:
398:. The third property then corresponds to the order of a sum being the order of the larger term, unless the two terms have the same order, in which case they may cancel and the sum may have larger order.
851:
500:
2210:
1909:
1795:
1690:
1946:
2863:
1830:
2274:
1751:
1339:
1304:
2239:
1852:
428:
2493:
2141:
3283:
1878:
965:
our "valuation" (satisfying the ultrametric inequality) is called an "exponential valuation" or "non-Archimedean absolute value" or "ultrametric absolute value";
2307:
1972:
506:, and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form a
2667:
2869:
3542:
3390:
is isomorphic to a subgroup of the real numbers under addition, but non-Archimedean ordered groups exist, such as the additive group of a
522:
to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.
3579:
3499:
3466:
1404:
809:
3608:
3567:
3534:
3458:
3391:
961:
In this article, we use the terms defined above, in the additive notation. However, some authors use alternative terms:
437:
3668:
3432:
3663:
3603:
2650:
1479:
968:
our "absolute value" (satisfying the triangle inequality) is called a "valuation" or an "Archimedean absolute value".
1658:
60:
3329:
2988:), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the
2146:
1514:) = (the degree of the extension of residue fields). It is also less than or equal to the degree of
1142:
803:
535:
3086:∪ {0} is the set of non-negative real numbers under multiplication. Then we say that the valuation is
2792:
541:
64:
56:
1883:
1756:
1664:
3028:
2626:
1914:
1274:
44:
36:
2826:
2572:
2093:
1800:
1623:
760:
431:
124:
3368:
of the order of the leading order term, but with the max convention it can be interpreted as the order.
2250:
2216:
1725:
1635:
1611:
1388:
1251:
111:
3319:
2642:
1531:
1435:
1319:
1284:
768:
372:
104:
52:
48:
2222:
1835:
411:
3559:
3314:
2634:
2465:
1563:
752:(Note that the directions of the inequalities are reversed from those in the additive notation.)
395:
388:
364:
76:
32:
2102:
51:
that provides a measure of the size or multiplicity of elements of the field. It generalizes to
3598:
3249:
3636:
3575:
3538:
3495:
3462:
3387:
2599:
1265:
1204:
764:
511:
384:
198:
80:
1638:, as in the examples below, and different valuations can define different completion fields.
3548:
3505:
3472:
3438:
3008:
1706:
1419:
1279:
84:
68:
3090:
if its range (the valuation group) is infinite (and hence has an accumulation point at 0).
3571:
3552:
3509:
3483:
3476:
1361:
1114:
1857:
2588:
2277:
1313:
1042:
3657:
3526:
3522:
3491:
1654:
1642:
1307:
1098:
376:
121:
72:
2455:{\displaystyle f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}}
2045:{\displaystyle \nu _{p}(a)=\max\{e\in \mathbb {Z} \mid p^{e}{\text{ divides }}a\};}
1610:, the associated valuation is equivalent to an absolute value, and hence induces a
2603:
406:
3639:
17:
3628:
3618:
3429:
3403:
In the tropical semiring, minimum and addition of real numbers are considered
3324:
531:
3644:
2766:{\displaystyle a=\pi ^{e_{a}}p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{n}^{e_{n}}}
135:
3614:
2970:{\displaystyle v_{\pi }(a/b)=e_{a}-e_{b},{\text{ for }}a,b\in R,a,b\neq 0.}
3364:
With the min convention here, the valuation is rather interpreted as the
2984:
such that (π') = (π) (that is, they generate the same ideal in
1607:
883:
507:
3351:, with no other meaning. Its properties are simply defined by the given
3514:
806:. In this case, we may pass to the additive notation with value group
28:
3457:, Mathematical Surveys and Monographs, vol. 124, Providence, RI:
3050:. The construction of the previous section applied to the prime ideal
3624:
3352:
1306:
these are precisely the equivalence classes of valuations for the
87:
in algebraic geometry. A field with a valuation on it is called a
2606:, with valuation in all cases returning the smallest exponent of
1261:
are equivalent if and only if they have the same valuation ring.
55:
the notion of size inherent in consideration of the degree of a
914:" satisfying the required properties, we can define valuation
71:, the degree of divisibility of a number by a prime number in
1634:
is not complete, one can use the valuation to construct its
555:
to Γ, with the ordering and group law extended by the rules
387:
applications, the first property implies that any non-empty
3046:, is a principal ideal domain whose field of fractions is
1278:
gives a complete classification of places of the field of
391:
of an analytic variety near a point contains that point.
1657:
as a uniform space. There is a related property known as
977:
There are several objects defined from a given valuation
3537:, vol. 29, New York, Heidelberg: Springer-Verlag,
3226:
are vector spaces over a non-discrete valuation field
3252:
2872:
2829:
2670:
2468:
2310:
2253:
2225:
2149:
2105:
1975:
1917:
1886:
1860:
1838:
1803:
1759:
1728:
1667:
1322:
1287:
846:{\displaystyle \Gamma _{+}\subseteq (\mathbb {R} ,+)}
812:
440:
414:
394:
The valuation can be interpreted as the order of the
363:
The second property asserts that any valuation is a
3301:will be radial under the additional condition that
1403:. The set of all such extensions is studied in the
495:{\displaystyle \min(a,+\infty )=\min(+\infty ,a)=a}
3277:
2969:
2857:
2765:
2487:
2454:
2268:
2233:
2204:
2135:
2044:
1940:
1903:
1872:
1846:
1824:
1789:
1745:
1684:
1333:
1298:
845:
494:
422:
3486:(1989) , "Valuations: paragraph 6 of chapter 9",
1626:with respect to this metric, then it is called a
3199:is invariant under arbitrary intersections. The
3171:are invariant under finite intersection. Also,
1998:
763:under multiplication, the last condition is the
465:
441:
430:in which case ∞ can be interpreted as +∞ in the
217:that satisfies the following properties for all
2563:consists of rational functions with no pole at
2292:(x), the rational functions on the affine line
1880:. The valuation group is the additive integers
3377:Again, swapped since using minimum convention.
3207:is the intersection of all circled subsets of
943:}, with multiplication and ordering based on
8:
2587:)). This can be generalized to the field of
2036:
2001:
646:satisfying the following properties for all
3517:written by one of the leading contributors.
3007:The previous example can be generalized to
1653:is called a complete valued field if it is
526:Multiplicative notation and absolute values
2649:. Since every principal ideal domain is a
2602:(its Cauchy completion), and the field of
3257:
3251:
2929:
2920:
2907:
2889:
2877:
2871:
2834:
2828:
2755:
2750:
2745:
2730:
2725:
2720:
2708:
2703:
2698:
2686:
2681:
2669:
2661:can be written (essentially) uniquely as
2473:
2467:
2446:
2434:
2422:
2397:
2385:
2367:
2354:
2342:
2330:
2309:
2260:
2256:
2255:
2252:
2227:
2226:
2224:
2205:{\displaystyle |a|_{p}:=p^{-\nu _{p}(a)}}
2185:
2177:
2164:
2159:
2150:
2148:
2124:
2109:
2104:
2092:Writing this multiplicatively yields the
2028:
2022:
2011:
2010:
1980:
1974:
1931:
1930:
1916:
1894:
1893:
1885:
1859:
1840:
1839:
1837:
1810:
1806:
1805:
1802:
1772:
1768:
1767:
1758:
1736:
1735:
1727:
1675:
1674:
1666:
1324:
1323:
1321:
1289:
1288:
1286:
830:
829:
817:
811:
439:
416:
415:
413:
371:. The third property is a version of the
2980:If π' is another irreducible element of
2621:Generalizing the previous examples, let
551:Instead of ∞, we adjoin a formal symbol
3422:
3347:The symbol ∞ denotes an element not in
3340:
518:is almost a semiring homomorphism from
99:One starts with the following objects:
1661:: it is equivalent to completeness if
1904:{\displaystyle \Gamma =\mathbb {Z} .}
1790:{\displaystyle R=\mathbb {Z} _{(p)},}
1685:{\displaystyle \Gamma =\mathbb {Z} ,}
1268:of valuations of a field is called a
7:
3411:; these are the semiring operations.
3003:-adic valuation on a Dedekind domain
2780:s are non-negative integers and the
1941:{\displaystyle a\in R=\mathbb {Z} ,}
1159:under the equivalence defined below.
1097:) > 0 (it is in fact a
379:adapted to an arbitrary Γ (see
3078:Vector spaces over valuation fields
2858:{\displaystyle v_{\pi }(0)=\infty }
2099:, which conventionally has as base
767:inequality, a stronger form of the
3451:Valuations, orderings, and Milnor
2852:
1887:
1825:{\displaystyle \mathbb {Z} _{(p)}}
1668:
1641:In general, a valuation induces a
814:
474:
456:
25:
3246:is circled or radial then so is
2508:) = k, the order of vanishing at
1405:ramification theory of valuations
75:, and the geometrical concept of
3195:. The set of circled subsets of
3019:its field of fractions, and let
2269:{\displaystyle \mathbb {Q} _{p}}
1203:if there is an order-preserving
3163:absorbs every finite subset of
1958:) measures the divisibility of
1746:{\displaystyle K=\mathbb {Q} ,}
1602:When the ordered abelian group
1199:, respectively, are said to be
882:defines a corresponding linear
405:is an additive subgroup of the
383:below). For valuations used in
3531:Commutative algebra, Volume II
3272:
3266:
2897:
2883:
2846:
2840:
2443:
2428:
2394:
2379:
2351:
2336:
2320:
2314:
2197:
2191:
2160:
2151:
1992:
1986:
1867:
1861:
1817:
1811:
1779:
1773:
1718:associated to a prime integer
1705:The most basic example is the
1458:) = , is called the
1024:is usually surjective so that
840:
826:
483:
468:
459:
444:
1:
3535:Graduate Texts in Mathematics
3459:American Mathematical Society
3392:non-Archimedean ordered field
3023:be a non-zero prime ideal of
2799:. In particular, the integer
1606:is the additive group of the
1334:{\displaystyle \mathbb {Q} .}
1299:{\displaystyle \mathbb {Q} :}
530:The concept was developed by
2787:are irreducible elements of
2598:}} (fractional powers), the
2571:, and the completion is the
2304:∈ X. For a polynomial
2234:{\displaystyle \mathbb {Q} }
1847:{\displaystyle \mathbb {Z} }
423:{\displaystyle \mathbb {R} }
3604:Encyclopedia of Mathematics
2651:unique factorization domain
2559:). Then the valuation ring
2488:{\displaystyle a_{k}\neq 0}
3685:
3490:(2nd ed.), New York:
2806:is uniquely determined by
2136:{\displaystyle 1/p=p^{-1}}
1722:, on the rational numbers
1460:reduced ramification index
3570:. Vol. 3. New York:
3564:Topological Vector Spaces
3492:W. H. Freeman and Company
3278:{\displaystyle f^{-1}(B)}
2653:, every non-zero element
2610:appearing in the series.
1692:but stronger in general.
1169:Equivalence of valuations
3330:Absolute value (algebra)
910:. Conversely, given a "
348:) = 0 for all
142:are extended to the set
3597:Danilov, V.I. (2001) ,
3409:tropical multiplication
3097:is a vector space over
2992:-adic valuation, where
1832:is the localization of
1344:Extension of valuations
542:multiplicative notation
401:For many applications,
381:Multiplicative notation
37:algebraic number theory
3562:; Wolff, M.P. (1999).
3437:, pages 47 to 49, via
3289:is circled then so is
3279:
3015:be a Dedekind domain,
2971:
2859:
2767:
2627:principal ideal domain
2489:
2456:
2270:
2235:
2206:
2137:
2046:
1942:
1905:
1874:
1848:
1826:
1791:
1747:
1686:
1659:spherical completeness
1598:Complete valued fields
1335:
1300:
1191:with valuation group Γ
847:
496:
424:
3280:
3242:be a linear map. If
2972:
2860:
2768:
2573:formal Laurent series
2490:
2457:
2271:
2236:
2207:
2138:
2055:and for a fraction, ν
2047:
1943:
1906:
1875:
1849:
1827:
1792:
1748:
1687:
1628:complete valued field
1336:
1301:
848:
761:positive real numbers
759:is a subgroup of the
540:writing the group in
497:
432:extended real numbers
425:
125:totally ordered group
3250:
3167:. Radial subsets of
2870:
2827:
2815:π-adic valuation of
2668:
2466:
2308:
2251:
2223:
2147:
2103:
2097:-adic absolute value
1973:
1915:
1884:
1858:
1836:
1801:
1757:
1753:with valuation ring
1726:
1665:
1478:) ≤ (the
1320:
1285:
1252:equivalence relation
810:
502:for any real number
438:
412:
112:multiplicative group
3669:Field (mathematics)
3560:Schaefer, Helmut H.
3513:. A masterpiece on
3449:Efrat, Ido (2006),
3320:Euclidean valuation
3113:. Then we say that
3069:-adic valuation of
2762:
2737:
2715:
2643:irreducible element
2300:, and take a point
2030: divides
1873:{\displaystyle (p)}
1854:at the prime ideal
1546:is defined to be e(
1426:be an extension of
1275:Ostrowski's theorem
769:triangle inequality
732:, with equality if
373:triangle inequality
312:, with equality if
53:commutative algebra
3664:Algebraic geometry
3637:Weisstein, Eric W.
3615:Discrete valuation
3574:. pp. 10–11.
3315:Discrete valuation
3275:
3123:if there exists a
2996: = (π).
2967:
2855:
2763:
2741:
2716:
2694:
2635:field of fractions
2485:
2452:
2284:Order of vanishing
2266:
2231:
2202:
2133:
2042:
1938:
1901:
1870:
1844:
1822:
1787:
1743:
1682:
1564:inseparable degree
1536:ramification index
1352:be a valuation of
1331:
1296:
1257:Two valuations of
973:Associated objects
876:Each valuation on
843:
514:, and a valuation
492:
420:
396:leading-order term
365:group homomorphism
356:, otherwise it is
85:analytic varieties
33:algebraic geometry
31:(in particular in
3544:978-0-387-90171-8
3434:Geometric Algebra
3405:tropical addition
3388:Archimedean group
2932:
2820:is then given by
2600:Levi-Civita field
2241:with respect to ν
2031:
1643:uniform structure
1566:of the extension
1502:is defined to be
1482:of the extension
1470:. It satisfies e(
1379:) is a valuation
1266:equivalence class
1205:group isomorphism
1016:), a subgroup of
537:Geometric Algebra
512:tropical semiring
510:, called the min
134:The ordering and
16:(Redirected from
3676:
3650:
3649:
3611:
3585:
3555:
3512:
3488:Basic algebra II
3484:Jacobson, Nathan
3479:
3441:
3439:Internet Archive
3427:
3412:
3401:
3395:
3384:
3378:
3375:
3369:
3362:
3356:
3350:
3345:
3284:
3282:
3281:
3276:
3265:
3264:
3085:
3072:
3068:
3049:
3034:
3026:
3018:
3014:
3009:Dedekind domains
2983:
2976:
2974:
2973:
2968:
2933:
2930:
2925:
2924:
2912:
2911:
2893:
2882:
2881:
2864:
2862:
2861:
2856:
2839:
2838:
2798:
2790:
2772:
2770:
2769:
2764:
2761:
2760:
2759:
2749:
2736:
2735:
2734:
2724:
2714:
2713:
2712:
2702:
2693:
2692:
2691:
2690:
2660:
2648:
2640:
2632:
2624:
2616:
2494:
2492:
2491:
2486:
2478:
2477:
2461:
2459:
2458:
2453:
2451:
2450:
2438:
2427:
2426:
2408:
2407:
2389:
2378:
2377:
2359:
2358:
2346:
2335:
2334:
2275:
2273:
2272:
2267:
2265:
2264:
2259:
2240:
2238:
2237:
2232:
2230:
2211:
2209:
2208:
2203:
2201:
2200:
2190:
2189:
2169:
2168:
2163:
2154:
2142:
2140:
2139:
2134:
2132:
2131:
2113:
2096:
2051:
2049:
2048:
2043:
2032:
2029:
2027:
2026:
2014:
1985:
1984:
1947:
1945:
1944:
1939:
1934:
1910:
1908:
1907:
1902:
1897:
1879:
1877:
1876:
1871:
1853:
1851:
1850:
1845:
1843:
1831:
1829:
1828:
1823:
1821:
1820:
1809:
1796:
1794:
1793:
1788:
1783:
1782:
1771:
1752:
1750:
1749:
1744:
1739:
1709:
1701:p-adic valuation
1691:
1689:
1688:
1683:
1678:
1652:
1648:
1621:
1617:
1605:
1420:finite extension
1398:
1367:
1355:
1340:
1338:
1337:
1332:
1327:
1305:
1303:
1302:
1297:
1292:
1280:rational numbers
1231:) = φ(
1219:
1190:
1164:Basic properties
1150:
1068:) ≥ 0,
1059:
1036:
1032:
1019:
1007:
987:
952:
948:
942:
913:
909:
881:
872:
852:
850:
849:
844:
833:
822:
821:
801:
792:
758:
747:
731:
705:
680:
673:
642:
621:
607:
603:
599:
575:
571:
567:
547:
501:
499:
498:
493:
429:
427:
426:
421:
419:
404:
311:
278:
249:
242:
213:
195:
185:
181:
177:
163:
159:
155:
145:
141:
129:
109:
69:complex analysis
21:
3684:
3683:
3679:
3678:
3677:
3675:
3674:
3673:
3654:
3653:
3635:
3634:
3596:
3593:
3588:
3582:
3572:Springer-Verlag
3558:
3545:
3521:
3502:
3482:
3469:
3448:
3444:
3428:
3424:
3420:
3415:
3402:
3398:
3385:
3381:
3376:
3372:
3363:
3359:
3348:
3346:
3342:
3338:
3311:
3305:is surjective.
3297:is radial then
3253:
3248:
3247:
3109:are subsets of
3083:
3080:
3070:
3066:
3062:
3055:
3047:
3044:
3032:
3024:
3016:
3012:
3005:
2981:
2931: for
2916:
2903:
2873:
2868:
2867:
2830:
2825:
2824:
2804:
2796:
2788:
2785:
2751:
2726:
2704:
2682:
2677:
2666:
2665:
2658:
2646:
2638:
2630:
2622:
2619:
2617:-adic valuation
2614:
2554:
2541:
2524:
2503:
2469:
2464:
2463:
2442:
2418:
2393:
2363:
2350:
2326:
2306:
2305:
2286:
2254:
2249:
2248:
2246:
2221:
2220:
2181:
2173:
2158:
2145:
2144:
2120:
2101:
2100:
2094:
2084:
2074:
2060:
2018:
1976:
1971:
1970:
1953:
1948:the valuation ν
1913:
1912:
1911:For an integer
1882:
1881:
1856:
1855:
1834:
1833:
1804:
1799:
1798:
1766:
1755:
1754:
1724:
1723:
1717:
1710:-adic valuation
1707:
1703:
1698:
1663:
1662:
1650:
1646:
1619:
1615:
1603:
1600:
1592:
1585:
1578:
1571:
1492:relative degree
1449:
1443:
1396:
1365:
1362:field extension
1353:
1346:
1318:
1317:
1283:
1282:
1237:
1226:
1218:
1214:
1207:
1198:
1194:
1188:
1186:
1179:
1173:Two valuations
1171:
1166:
1155:, the class of
1148:
1136:
1129:
1122:
1106:
1079:
1057:
1050:
1034:
1031:
1025:
1017:
1006:
1000:
998:valuation group
978:
975:
959:
950:
944:
920:
915:
911:
907:
900:
887:
877:
870:
865:) = −log
860:
854:
813:
808:
807:
799:
796:| ⋅ |
794:
790:
783:
776:
773:|a+b|
771:
756:
745:
738:
733:
728:
721:
714:
711:|a+b|
709:
703:
696:
689:
684:
675:
674:if and only if
667:
662:
631:
628:| ⋅ |
626:
617:
605:
601:
579:
573:
569:
559:
545:
528:
436:
435:
410:
409:
402:
282:
253:
244:
243:if and only if
233:
204:
193:
183:
179:
176:+ ∞ = ∞ + ∞ = ∞
167:
161:
157:
150:
146:} by the rules
143:
139:
127:
107:
97:
23:
22:
18:Krull valuation
15:
12:
11:
5:
3682:
3680:
3672:
3671:
3666:
3656:
3655:
3652:
3651:
3632:
3622:
3612:
3592:
3591:External links
3589:
3587:
3586:
3580:
3556:
3543:
3527:Samuel, Pierre
3523:Zariski, Oscar
3520:Chapter VI of
3518:
3500:
3480:
3467:
3445:
3443:
3442:
3421:
3419:
3416:
3414:
3413:
3396:
3379:
3370:
3357:
3339:
3337:
3334:
3333:
3332:
3327:
3322:
3317:
3310:
3307:
3274:
3271:
3268:
3263:
3260:
3256:
3240:f : X → Y
3079:
3076:
3060:
3053:
3042:
3004:
2998:
2978:
2977:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2928:
2923:
2919:
2915:
2910:
2906:
2902:
2899:
2896:
2892:
2888:
2885:
2880:
2876:
2865:
2854:
2851:
2848:
2845:
2842:
2837:
2833:
2802:
2783:
2774:
2773:
2758:
2754:
2748:
2744:
2740:
2733:
2729:
2723:
2719:
2711:
2707:
2701:
2697:
2689:
2685:
2680:
2676:
2673:
2618:
2612:
2589:Puiseux series
2550:
2537:
2520:
2499:
2484:
2481:
2476:
2472:
2449:
2445:
2441:
2437:
2433:
2430:
2425:
2421:
2417:
2414:
2411:
2406:
2403:
2400:
2396:
2392:
2388:
2384:
2381:
2376:
2373:
2370:
2366:
2362:
2357:
2353:
2349:
2345:
2341:
2338:
2333:
2329:
2325:
2322:
2319:
2316:
2313:
2285:
2282:
2278:p-adic numbers
2263:
2258:
2242:
2229:
2199:
2196:
2193:
2188:
2184:
2180:
2176:
2172:
2167:
2162:
2157:
2153:
2130:
2127:
2123:
2119:
2116:
2112:
2108:
2080:
2070:
2056:
2053:
2052:
2041:
2038:
2035:
2025:
2021:
2017:
2013:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1983:
1979:
1949:
1937:
1933:
1929:
1926:
1923:
1920:
1900:
1896:
1892:
1889:
1869:
1866:
1863:
1842:
1819:
1816:
1813:
1808:
1786:
1781:
1778:
1775:
1770:
1765:
1762:
1742:
1738:
1734:
1731:
1713:
1702:
1699:
1697:
1694:
1681:
1677:
1673:
1670:
1599:
1596:
1590:
1583:
1576:
1569:
1445:
1439:
1387:such that the
1345:
1342:
1330:
1326:
1295:
1291:
1235:
1224:
1216:
1212:
1196:
1192:
1184:
1177:
1170:
1167:
1165:
1162:
1161:
1160:
1151:associated to
1139:
1134:
1127:
1120:
1109:
1104:
1081:is the set of
1077:
1069:
1052:is the set of
1048:
1043:valuation ring
1038:
1027:
1002:
974:
971:
970:
969:
966:
958:
955:
918:
905:
898:
868:
858:
842:
839:
836:
832:
828:
825:
820:
816:
804:absolute value
797:
788:
781:
774:
750:
749:
743:
736:
726:
719:
712:
707:
701:
694:
687:
686:|ab|
682:
665:
644:
643:
629:
610:
609:
577:
527:
524:
491:
488:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
446:
443:
418:
330:
329:
280:
251:
215:
214:
188:
187:
165:
132:
131:
118:
96:
93:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3681:
3670:
3667:
3665:
3662:
3661:
3659:
3647:
3646:
3641:
3638:
3633:
3630:
3626:
3623:
3620:
3616:
3613:
3610:
3606:
3605:
3600:
3595:
3594:
3590:
3583:
3581:9780387987262
3577:
3573:
3569:
3565:
3561:
3557:
3554:
3550:
3546:
3540:
3536:
3532:
3528:
3524:
3519:
3516:
3511:
3507:
3503:
3501:0-7167-1933-9
3497:
3493:
3489:
3485:
3481:
3478:
3474:
3470:
3468:0-8218-4041-X
3464:
3460:
3456:
3452:
3447:
3446:
3440:
3436:
3435:
3431:
3426:
3423:
3417:
3410:
3406:
3400:
3397:
3393:
3389:
3383:
3380:
3374:
3371:
3367:
3361:
3358:
3354:
3344:
3341:
3335:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3312:
3308:
3306:
3304:
3300:
3296:
3292:
3288:
3269:
3261:
3258:
3254:
3245:
3241:
3237:
3233:
3229:
3225:
3221:
3218:Suppose that
3216:
3214:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3150:
3146:
3143:implies that
3142:
3138:
3134:
3130:
3126:
3122:
3121:
3117:
3112:
3108:
3104:
3100:
3096:
3093:Suppose that
3091:
3089:
3082:Suppose that
3077:
3075:
3073:
3063:
3056:
3045:
3038:
3030:
3022:
3010:
3002:
2999:
2997:
2995:
2991:
2987:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2926:
2921:
2917:
2913:
2908:
2904:
2900:
2894:
2890:
2886:
2878:
2874:
2866:
2849:
2843:
2835:
2831:
2823:
2822:
2821:
2819:
2818:
2811:
2809:
2805:
2794:
2791:that are not
2786:
2779:
2756:
2752:
2746:
2742:
2738:
2731:
2727:
2721:
2717:
2709:
2705:
2699:
2695:
2687:
2683:
2678:
2674:
2671:
2664:
2663:
2662:
2656:
2652:
2644:
2636:
2628:
2613:
2611:
2609:
2605:
2601:
2597:
2593:
2590:
2586:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2553:
2549:
2545:
2540:
2536:
2532:
2528:
2523:
2519:
2515:
2511:
2507:
2502:
2498:
2482:
2479:
2474:
2470:
2447:
2439:
2435:
2431:
2423:
2419:
2415:
2412:
2409:
2404:
2401:
2398:
2390:
2386:
2382:
2374:
2371:
2368:
2364:
2360:
2355:
2347:
2343:
2339:
2331:
2327:
2323:
2317:
2311:
2303:
2299:
2295:
2291:
2283:
2281:
2279:
2261:
2247:is the field
2245:
2218:
2213:
2194:
2186:
2182:
2178:
2174:
2170:
2165:
2155:
2128:
2125:
2121:
2117:
2114:
2110:
2106:
2098:
2090:
2088:
2083:
2078:
2073:
2068:
2064:
2059:
2039:
2033:
2023:
2019:
2015:
2007:
2004:
1995:
1989:
1981:
1977:
1969:
1968:
1967:
1965:
1962:by powers of
1961:
1957:
1952:
1935:
1927:
1924:
1921:
1918:
1898:
1890:
1864:
1814:
1784:
1776:
1763:
1760:
1740:
1732:
1729:
1721:
1716:
1711:
1700:
1695:
1693:
1679:
1671:
1660:
1656:
1644:
1639:
1637:
1633:
1629:
1625:
1614:on the field
1613:
1609:
1597:
1595:
1593:
1586:
1579:
1572:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1448:
1442:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1408:
1406:
1402:
1394:
1390:
1386:
1382:
1378:
1374:
1373:
1370:extension of
1363:
1359:
1351:
1343:
1341:
1328:
1315:
1312:
1310:
1293:
1281:
1277:
1276:
1271:
1267:
1262:
1260:
1255:
1253:
1250:. This is an
1249:
1245:
1241:
1234:
1230:
1223:
1210:
1206:
1202:
1183:
1176:
1168:
1163:
1158:
1154:
1146:
1145:
1140:
1137:
1130:
1123:
1117:
1116:
1110:
1107:
1100:
1099:maximal ideal
1096:
1092:
1088:
1084:
1080:
1074:
1070:
1067:
1063:
1055:
1051:
1045:
1044:
1039:
1030:
1023:
1015:
1011:
1005:
999:
995:
991:
990:
989:
985:
981:
972:
967:
964:
963:
962:
956:
954:
947:
941:
937:
933:
929:
925:
921:
917:|a|
908:
904:|b|
901:
897:|a|
894:
890:
885:
880:
874:
871:
867:|a|
864:
857:
837:
834:
823:
818:
805:
800:
791:
787:|b|
784:
780:|a|
777:
770:
766:
762:
753:
746:
742:|b|
739:
735:|a|
729:
725:|b|
722:
718:|a|
715:
708:
704:
700:|b|
697:
693:|a|
690:
683:
678:
672:
668:
664:|a|
661:
660:
659:
657:
653:
649:
640:
636:
632:
625:
624:
623:
620:
615:
598:
594:
590:
586:
582:
578:
566:
562:
558:
557:
556:
554:
549:
543:
539:
538:
533:
525:
523:
521:
517:
513:
509:
505:
489:
486:
480:
477:
471:
462:
453:
450:
447:
433:
408:
399:
397:
392:
390:
386:
382:
378:
377:metric spaces
374:
370:
366:
361:
359:
355:
351:
347:
343:
339:
335:
327:
323:
319:
315:
309:
305:
301:
297:
293:
289:
285:
281:
276:
272:
268:
264:
260:
256:
252:
247:
240:
236:
232:
231:
230:
228:
224:
220:
211:
207:
203:
202:
201:
200:
196:
192:valuation of
175:
171:
166:
154:
149:
148:
147:
137:
126:
123:
119:
116:
113:
106:
102:
101:
100:
94:
92:
90:
86:
82:
78:
74:
73:number theory
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
3643:
3602:
3563:
3530:
3487:
3454:
3450:
3433:
3425:
3408:
3404:
3399:
3382:
3373:
3365:
3360:
3343:
3302:
3298:
3294:
3290:
3286:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3217:
3212:
3208:
3204:
3201:circled hull
3200:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3124:
3119:
3115:
3114:
3110:
3106:
3102:
3098:
3094:
3092:
3088:non-discrete
3087:
3081:
3065:
3058:
3051:
3040:
3036:
3029:localization
3027:. Then, the
3020:
3006:
3000:
2993:
2989:
2985:
2979:
2816:
2814:
2812:
2807:
2800:
2781:
2777:
2775:
2654:
2620:
2607:
2595:
2591:
2584:
2580:
2576:
2568:
2564:
2560:
2556:
2551:
2547:
2543:
2538:
2534:
2530:
2526:
2521:
2517:
2513:
2509:
2505:
2500:
2496:
2301:
2297:
2293:
2289:
2287:
2243:
2214:
2091:
2086:
2081:
2076:
2071:
2066:
2062:
2057:
2054:
1963:
1959:
1955:
1950:
1719:
1714:
1704:
1640:
1631:
1627:
1601:
1588:
1581:
1574:
1567:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1446:
1440:
1431:
1427:
1423:
1415:
1411:
1409:
1400:
1392:
1384:
1380:
1376:
1371:
1369:
1357:
1349:
1347:
1308:
1273:
1269:
1263:
1258:
1256:
1247:
1243:
1239:
1232:
1228:
1221:
1208:
1200:
1181:
1174:
1172:
1156:
1152:
1143:
1132:
1125:
1118:
1112:
1102:
1094:
1090:
1086:
1082:
1075:
1072:
1065:
1061:
1053:
1046:
1041:
1028:
1021:
1013:
1009:
1003:
997:
993:
983:
979:
976:
960:
945:
939:
935:
931:
927:
923:
916:
903:
896:
892:
888:
878:
875:
866:
862:
855:
795:
786:
779:
772:
754:
751:
741:
734:
724:
717:
710:
699:
692:
685:
676:
670:
663:
655:
651:
647:
645:
638:
634:
627:
618:
613:
611:
596:
592:
588:
584:
580:
564:
560:
552:
550:
536:
534:in his book
529:
519:
515:
503:
434:; note that
407:real numbers
400:
393:
380:
368:
362:
357:
353:
349:
345:
341:
337:
333:
332:A valuation
331:
325:
321:
317:
313:
307:
303:
299:
295:
291:
287:
283:
274:
270:
266:
262:
258:
254:
245:
238:
234:
226:
222:
218:
216:
209:
205:
191:
189:
173:
169:
152:
133:
114:
98:
89:valued field
88:
79:between two
61:multiplicity
40:
26:
3640:"Valuation"
3599:"Valuation"
3211:containing
3064:yields the
2604:Hahn series
2079:) − ν
1389:restriction
1314:completions
1242:)) for all
1073:prime ideal
994:value group
957:Terminology
765:ultrametric
622:is any map
358:non-trivial
212:→ Γ ∪ {∞}
3658:Categories
3629:PlanetMath
3619:PlanetMath
3553:0322.13001
3510:0694.16001
3477:1103.12002
3430:Emil Artin
3418:References
3325:Field norm
3238:, and let
3175:is called
3151:is called
3131:such that
3039:, denoted
2793:associates
2776:where the
2546:) −
2217:completion
1636:completion
1220:such that
1201:equivalent
986:→ Γ ∪ {∞}
853:by taking
532:Emil Artin
95:Definition
3645:MathWorld
3625:Valuation
3609:EMS Press
3529:(1976) ,
3259:−
3189:|λ| ≥ |α|
3157:absorbing
3141:|λ| ≥ |α|
3101:and that
2962:≠
2944:∈
2914:−
2879:π
2853:∞
2836:π
2739:⋯
2679:π
2495:, define
2480:≠
2436:−
2413:⋯
2387:−
2344:−
2183:ν
2179:−
2126:−
2016:∣
2008:∈
1978:ν
1922:∈
1888:Γ
1669:Γ
1532:separable
1211: : Γ
824:⊆
815:Γ
614:valuation
572:∈
546:(Γ, ·, ≥)
475:∞
457:∞
385:geometric
182:∈
136:group law
128:(Γ, +, ≥)
81:algebraic
41:valuation
3366:negative
3309:See also
3191:implies
3135:∈
3127:∈
3118:absorbs
2288:Let K =
1696:Examples
1655:complete
1624:complete
1608:integers
1558:, where
1422:and let
1356:and let
1113:residue
1085:∈
1056:∈
1020:(though
982: :
884:preorder
654:∈
637:→ Γ ∪ {
633: :
604:∈
600:for all
568:for all
508:semiring
294:) ≥ min(
208: :
178:for all
160:∈
156:for all
110:and its
45:function
3515:algebra
3455:-theory
3293:but if
3193:λ A ⊆ A
3177:circled
3145:B ⊆ λ A
2633:be its
2583:−
1797:where
1562:is the
1522:. When
1490:). The
612:Then a
338:trivial
197:is any
190:Then a
122:abelian
77:contact
29:algebra
3578:
3551:
3541:
3508:
3498:
3475:
3465:
3386:Every
3353:axioms
3230:, let
3153:radial
3011:. Let
2641:be an
2637:, and
2516:; and
1649:, and
1612:metric
1534:, the
1480:degree
1434:. The
802:is an
793:, and
757:Γ
716:≤ max(
144:Γ ∪ {∞
3336:Notes
3285:. If
3236:B ⊆ Y
3232:A ⊆ X
2625:be a
2575:ring
2462:with
2143:, so
2069:) = ν
1630:. If
1618:. If
1580:over
1542:over
1498:over
1466:over
1436:index
1418:be a
1368:. An
1360:be a
1311:-adic
1270:place
1195:and Γ
1144:place
1115:field
1089:with
1060:with
241:) = ∞
105:field
63:of a
49:field
47:on a
43:is a
39:), a
3576:ISBN
3539:ISBN
3496:ISBN
3463:ISBN
3407:and
3299:f(A)
3291:f(A)
3222:and
3187:and
3139:and
3105:and
2813:The
2533:) =
2215:The
1450:, e(
1444:in Γ
1438:of Γ
1410:Let
1375:(to
1348:Let
1180:and
1141:the
1111:the
1071:the
1040:the
992:the
949:and
755:If
389:germ
320:) ≠
269:) +
261:) =
168:∞ +
151:∞ ≥
65:zero
57:pole
3627:at
3617:at
3568:GTM
3549:Zbl
3506:Zbl
3473:Zbl
3203:of
3183:in
3179:if
3159:if
3155:or
3057:of
3035:at
3031:of
2795:of
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2645:of
2276:of
2219:of
2089:).
1999:max
1645:on
1622:is
1538:of
1530:is
1494:of
1462:of
1430:to
1399:is
1395:to
1391:of
1383:of
1364:of
1316:of
1264:An
1246:in
1215:→ Γ
1187:of
1147:of
1101:of
1033:=
996:or
922:= {
679:= 0
616:of
544:as
466:min
442:min
375:on
367:on
352:in
340:if
336:is
302:),
248:= 0
225:in
199:map
138:on
120:an
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67:in
59:or
35:or
27:In
3660::
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3607:,
3601:,
3566:.
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3504:,
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3215:.
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3074:.
3052:PR
2965:0.
2810:.
2778:e'
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2594:{{
2579:((
2567:=
2512:=
2296:=
2280:.
2212:.
2171::=
1966::
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1254:.
1124:=
1108:),
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1008:=
988:;
953:.
938:≼
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930:≼
926::
902:≤
895:⇔
891:≼
886::
873:.
785:+
778:≤
740:≠
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698:·
691:=
669:=
658::
650:,
641:}
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591:·
587:=
583:·
563:≤
548::
360:.
328:).
310:))
290:+
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221:,
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103:a
91:.
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3453:K
3394:.
3355:.
3349:Γ
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3295:A
3287:A
3273:)
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3267:(
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3129:K
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2935:a
2927:,
2922:b
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2898:)
2895:b
2891:/
2887:a
2884:(
2875:v
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2847:)
2844:0
2841:(
2832:v
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2808:a
2803:a
2801:e
2797:π
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2700:1
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2675:=
2672:a
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2429:(
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2410:+
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2399:k
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2321:)
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2315:(
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2302:a
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2257:Q
2244:p
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2198:)
2195:a
2192:(
2187:p
2175:p
2166:p
2161:|
2156:a
2152:|
2129:1
2122:p
2118:=
2115:p
2111:/
2107:1
2095:p
2087:b
2085:(
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2075:(
2072:p
2067:b
2065:/
2063:a
2061:(
2058:p
2040:;
2037:}
2034:a
2024:e
2020:p
2012:Z
2005:e
2002:{
1996:=
1993:)
1990:a
1987:(
1982:p
1964:p
1960:a
1956:a
1954:(
1951:p
1936:,
1932:Z
1928:=
1925:R
1919:a
1899:.
1895:Z
1891:=
1868:)
1865:p
1862:(
1841:Z
1818:)
1815:p
1812:(
1807:Z
1785:,
1780:)
1777:p
1774:(
1769:Z
1764:=
1761:R
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1737:Q
1733:=
1730:K
1720:p
1715:p
1712:ν
1708:p
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1676:Z
1672:=
1651:K
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1587:/
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1329:.
1325:Q
1309:p
1294::
1290:Q
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1238:(
1236:1
1233:v
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1227:(
1225:2
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1217:2
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1131:/
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1103:R
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1091:v
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1012:(
1010:v
1004:v
1001:Γ
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980:v
951:≼
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940:b
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924:b
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912:≼
906:v
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861:(
859:+
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838:+
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744:v
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677:a
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666:v
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608:.
606:Γ
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574:Γ
570:α
565:α
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553:O
520:K
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504:a
490:a
487:=
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481:a
478:,
472:+
469:(
463:=
460:)
454:+
451:,
448:a
445:(
417:R
403:Γ
369:K
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342:v
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184:Γ
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