240031 Proseminar "Quadratische Formen" (PS) (SoSe 2021)

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Zum Inhalt des Proseminars:
Eine quadratische Form über einem Körper K ist in geeigneten Koordinaten ein Ausdruck q(x) = a_1 x_1^2 + a_2 x_2^2 + ... + a_n x_n^2 mit a_1, a_2, ..., a_n in K. Über den reellen Zahlen werden quadratische Formen durch ihre Signatur klassifiziert. Die zugehörigen Quadriken {x ∈ R^n ; Q(x) = r} lassen sich geometrisch unter anderem durch den Satz von der Hauptachsentransformation beschreiben.

Das Proseminar wird die Arithmetik quadratischer Formen beleuchten. Ein Hauptziel ist das lokal-global Prinzip von Hasse und Minkowski: eine quadratische Form über Q hat eine Nullstelle, wenn sie überall lokal eine Nullstelle hat. Dies illustriert eine fundamentale Technik der Zahlentheorie, die Aspekte bezüglich einer fixierten Primzahl p durch Übergang zum lokalen Körper Q_p der p-adischen Zahlen isoliert zu betrachten. Die Aspekte der Positivität und Größe studiert man durch Übergang zum lokalen Körper R.

Was man lernt: quadratisches Reziprozitätsgesetz, p-adische Zahlen, Invarianten von quadratische Formen, Grothendieck Gruppen.

Vorbesprechung: Donnerstag 25.02.21, 10:15 Uhr, auf Zoom :
https://uni-bielefeld.zoom.us/j/2832073545?pwd=aUxadE8yTVJkLyt1Y3VZUEdOMHRZZz09
Meeting ID: 283 207 3545
Passcode: o4ik5jn3k

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Dates ( Calendar view )

Frequency Weekday Time Format / Place Period  
weekly Do 12-14 ONLINE   12.04.-23.07.2021

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Module Course Requirements  
24-B-GEO_ver1 Geometrie (Gym/Ge) Proseminar Study requirement
Ungraded examination
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24-B-PX Praxismodul Proseminar Study requirement
Ungraded examination
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24-E Ergänzungsmodul Mathematik Proseminar Study requirement
Ungraded examination
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Type(s) / SWS (hours per week per semester)
proseminar (PS) / 2
Department
Faculty of Mathematics
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253950698