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Dirichlet sai matematiikan opinnoilleen vanhempiensa taloudellisen tuen, vaikka he toivoivat pojastaan lakimiestä. Saksassa korkeamman matematiikan opiskelua olisi voinut jatkaa vain [[Carl Friedrich Gauss|Gaussin]] ohjauksessa Göttingenissä. Gauss oli virallisesti tähtitieteen professori eikä pitänyt opettamisesta, joten Dirichlet matkusti [[Pariisi|Pariisiin]] toukokuussa 1822. Siellä hän meni luennoille ja oppi matematiikkaa muun muassa[[Jean Nicolas Pierre Hachette|Hachettelta]]. Dirichlet opiskeli itsenäisesti Gaussin kirjaa ''[[Disquisitiones Arithmeticae]]'', jota hän piti aina mukanaan. Vuonna 1823 hän sai suositusten ansiosta töitä [[Maximilien Sebastien Foy|General Foylta]], joka palkkasi Dirichlet'n opettamaan lapsilleen saksaa. Nyt Dirichlet pärjäsi itsenäisesti ilman vanhempiensa taloudellista tukea.<ref name=James>{{cite book| last = James| first = Ioan Mackenzie | title=Remarkable Mathematicians: From Euler to von Neumann | year=2003| publisher=Cambridge University Press| location = | isbn= 978-0-521-52094-2 | pages= 103–109 }}</ref>
HisDirichlet first original research, comprising part of a proof oftodisti [[Fermat'sn lastsuuri theoremlause|Fermat'n suuren lauseen]] forosittain the casetapauksessa <math>n = 5</math>, brought him immediate fame, being the first advance in the theorem since. [[Pierre de Fermat|Fermat]]'s ownoli prooftodistanut of the casetapauksen <math>n = 4</math> andja [[Leonhard Euler|Euler]]'s proof fortapauksen <math>n = 3</math>. [[Adrien-Marie Legendre]], onemutta oftämän thejälkeen referees,lauseen soontodistus completedei theollut proofedistynyt. forSen this case;vuoksi Dirichlet'stä completedtuli hispian ownkuuluisa. proof[[Adrien-Marie aLegendre]] shortsai timetodistettua aftertapauksen Legendre,näihin andaikoihin aja fewDirichlet yearssaatti latertodistuksensa producedpian aloppuun. fullMuutamaa proofvuotta formyöhemmin theDirichlet todisti casetapauksen <math>n = 14</math>.<ref name=Krantz>{{cite book| last = Krantz| first = Steven | title=The Proof is in the Pudding: The Changing Nature of Mathematical Proof | year=2011| publisher=Springer| location = | isbn= 978-0-387-48908-7| pages= 55–58}}</ref> In JuneKesäkuussa 1825 hehänet waskutsuttiin accepted to lecture on his partial proof for theluennoimaan casetapauksen <math>n = 5</math> at thetodistuksesta [[French Academy of Sciences]],-oppilaitoksessa. anTämä exceptionaloli feat for aharvinaista 20-vuotiaalle year oldnuorelle, studentjolla withei noollut degreetutkintoa.<ref name=Elstrodt/> HisLuennon lecture at the Academy has also putmyötä Dirichlet in close contact withtutustui [[Joseph Fourier|FourierFourieriin]] andja [[Siméon Denis Poisson|PoissonPoissoniin]], whoja raisedsiten hisinnostui interestteoreettisesta in theoretical physicsfysiikasta, especiallyerityisesti Fourier'sn työstä [[Heat equationlämpöyhtälö|analytic theory oflämpöyhtälön heatparissa]].
===Back to PrussiaBreslau, BreslauPreussi (1825–28)===
As General Foy died inkuoli Novembermarraskuussa 1825 andja hemenetettyään could not find any paying position in France,työpaikkansa Dirichlet'n hadtäytyi topalata return to PrussiaPreussiin. Fourier andja Poisson introducedesittelivät him tohänet [[Alexander von Humboldt|Humboldtille]], who had been called to join the court of King [[Frederick William III of Prussia|Friedrich Wilhelm III]]. Humboldt, planningtuki toDirichlet'tä makemuun Berlinmuassa akirjoittamalla centersuosituskirjeen ofGaussille science and research, immediately offered his help toylistäen Dirichlet,'n sending letters in his favour to the Prussian government and to the [[Prussian Academy of Sciences]]. Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir onlahjakkuutta Fermat's theorem wrote with an unusual amount of praise that "Dirichlet showedn excellentlauseen talent"käsittelyssä.<ref name=Goldstein>{{cite book| last = Goldstein| first = Cathérine | coauthors = Catherine Goldstein, Norbert Schappacher, Joachim Schwermer| title=The shaping of arithmetic: after C.F. Gauss's Disquisitiones Arithmeticae| year=2007| publisher=Springer| location = | isbn= 978-3-540-20441-1| pages= 204–208}}</ref> With the support of Humboldt and Gauss, Dirichlet was offered a teaching position at the [[University of Wrocław|University of Breslau]] (now the University of Wrocław in [[Poland]]). However, as he had not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to the [[University of Bonn]]. Again his lack of fluency in Latin rendered him unable to hold the required public disputation of his thesis; after much discussion, the University decided to bypass the problem by awarding him a [[honorary doctorate]] in February 1827. Also, the Minister of education granted him a dispensation for the Latin disputation required for the [[Habilitation]]. Dirichlet earned the Habilitation and lectured in the 1827/28 year as a [[Privatdozent]] at Breslau.<ref name=Elstrodt/>
While in Breslau, Dirichlet continued his number theoretic research, publishing important contributions to the [[Quartic reciprocity|biquadratic reciprocity]] law which at the time was a focal point of Gauss' research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from [[Friedrich Bessel]], to arrange for him the desired transfer to Berlin. Given Dirichlet's young age (he was 23 years old at the time), Humboldt was only able to get him a trial position at the [[Prussian Military Academy]] in Berlin while remaining nominally employed by the University of Breslau. The probation was extended for three years until the position becoming definitive in 1831.
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