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Extra info for Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory

Example text

But Gauss himself had become the first mathematician to extend the notions of number theory itself beyond the rational "field" in which every sum, difference, product and (unlike among the natural numbers) quotient of two numbers is another number 1 where a in the field. He did this for those numbers of the form a b and b are rational numbers. These numbers also form a field, an algebraic and so on; and number field, as do the numbers of the form a+ b they are among the fields which are the subject of what is called algebraic number theory.

During these years the friendship between Hilbert and Minkowski deepened. Minkowski was a frequent vacation guest at Rauschen. " In their correspondence they continued to address each other by the formal pronoun "Sie"; but Hilbert, sending Minkowski a reprint of his first published work - the paper which Klein had presented to the Leipzig Academy the previous year- inscribed it: "To his friend and colleague in the closest sense ... " That first year as a docent, Hilbert made none of the trips which he had so optimistically planned in order to compensate himself for the isolation of Konigsberg.

There were no more calls and the copying of the habilitation paper had to be postponed. He managed only to drag himself to lectures and meetings. " He may also have been just a little homesick. By the end of June, on his way back to Konigsberg, he was happy and full of enthusiasm. He stopped in Gottingen and reported to Klein on the Paris experiences. It was his first visit to the University, and he found himself charmed by the little town and the pretty, billy countryside, so different from the bustling city of Konigsberg and the flat meadows beyond it.