Hendrik Lenstra

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Finite fields are finite, and they are fields, and as a result one can combine algebraic arguments with counting arguments in their study. This was illustrated in a lecture given at the 2009 April Fools’ meeting of the Leiden bachelor seminar. Here is the text of that lecture. INTRODUCTION. On a recent algebra test, I asked the students to factor the(More)
γ = c0 + c1p+ c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation(More)
In mijn Californische tijd gebeurde het een keer dat het gebied waar ik woonde en werkzaam was een ander netnummer kreeg: in plaats van 415 was het voortaan 510. Deze dramatische ingreep in het leven van miljoenen telefoongebruikers was voor een employé van de Oakland Tribune aanleiding ons Department of Mathematics te bellen met de vraag wat voor goeds er(More)
Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to(More)
iii Acknowledgements I would like to sincerely thank Owen Biesel, my Master's thesis advisor, who has been extremely kind and helpful with me. He generously and clearly explained to me his research interests, and he was always very patient and encouraging while assisting me with my thesis. I would also like to thank It was a pleasure to attend their(More)
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