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\n Let\n \n \n \n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n E\/K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an abelian extension of number fields, with\n \n \n \n \n E<\/mml:mi>\n \n \u2260\n \n <\/mml:mo>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n E \\ne \\Bbb Q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denote the absolute discriminant and degree of\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denote an element of the Galois group of\n \n \n \n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n E\/K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We prove the following theorems, assuming the Extended Riemann Hypothesis:\n <\/p>\n

\n There is a degree-\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n prime\n \n \n \n \n p<\/mml:mi>\n <\/mml:mrow>\n {\\frak p}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n (<\/mml:mo>\n \n \n p<\/mml:mi>\n <\/mml:mstyle>\n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n \\left (\\frac {\\displaystyle \\frak p}{E\/K}\\right ) =\\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , satisfying\n \n \n \n \n N<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n N{\\frak p}\\le (1+o(1))(\\log \\Delta +2n)^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n There is a degree-\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n prime\n \n \n \n p<\/mml:mi>\n \\frak p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n (<\/mml:mo>\n \n \n p<\/mml:mi>\n <\/mml:mstyle>\n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n )<\/mml:mo>\n <\/mml:mrow>\n \\left (\\frac {\\displaystyle \\frak p}{E\/K}\\right )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n generates the same group as\n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , satisfying\n \n \n \n \n N<\/mml:mi>\n \n p<\/mml:mi>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n N{\\frak p}\\le (1+o(1))(\\log \\Delta )^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n For\n \n \n \n \n K<\/mml:mi>\n =<\/mml:mo>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n K=\\Bbb Q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , there is a prime\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n (<\/mml:mo>\n \n \n p<\/mml:mi>\n <\/mml:mstyle>\n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mfrac>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n \\left (\\frac {\\displaystyle \\frak p}{E\/\\mathbb {Q}}\\right )=\\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , satisfying\n \n \n \n \n p<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n p\\le (1+o(1))(\\log \\Delta )^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n In (1) and (2) we can in fact take\n \n \n \n p<\/mml:mi>\n \\frak p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to be unramified in\n \n \n \n \n K<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n K\/\\Bbb Q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A special case of this result is the following.\n <\/p>\n

\n [(4)] If\n \n \n \n \n gcd<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \\gcd (m,q)=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , the least prime\n \n \n \n \n p<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n m<\/mml:mi>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n p\\equiv m\\pmod q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfies\n \n \n \n \n p<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n (<\/mml:mo>\n \n \u03c6\n \n <\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n q<\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n p\\le (1+o(1))(\\varphi (q)\\log q)^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>\n

\n It follows from our proof that (1)\u2013(3) also hold for arbitrary Galois extensions, provided we replace\n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \\sigma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by its conjugacy class\n \n \n \n \n \n \u27e8\n \n <\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n \n \u27e9\n \n <\/mml:mo>\n <\/mml:mrow>\n \\langle \\sigma \\rangle<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Our theorems lead to explicit versions of (1)\u2013(4), including the following: the least prime\n \n \n \n \n p<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n m<\/mml:mi>\n \n (<\/mml:mo>\n mod<\/mml:mi>\n \n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n p \\equiv m \\pmod q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is less than\n \n \n \n \n 2<\/mml:mn>\n (<\/mml:mo>\n q<\/mml:mi>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n q<\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n 2( q \\log q )^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-96-00763-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1717-1735","source":"Crossref","is-referenced-by-count":55,"title":["Explicit bounds for primes in residue classes"],"prefix":"10.1090","volume":"65","author":[{"given":"Eric","family":"Bach","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Sorenson","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"M. 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