Abstract: In this expository talk, I'll give some background on the Northcott and Bogomolov properties for field extensions of the rational numbers, possibly of infinite degree. This will involve defining the Weil height and talking about what bounds we expect to see in such fields. I'll survey some results of Bombieri-Zannier (nontrivial examples of property B), Fili-Miner (equdistribution and property N), and Amoroso-Dvornicich (property N in abelian fields). This talk will serve to provide some background for Preston's Candidacy Talk next week.
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