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Project: Math 582b
Views: 2495

Feb 8, 2016: Part 3 of course

William Stein

So far we have done 2 things:

Part 1: deep dive into arithmetic in number fields (and finite fields)

  • Showed the depth of functionality in Sage/Magma/libraries for just doing additional and multiplication of elements

  • Learned basics of Sage's api and why it is what it is.

  • Nontrivial: spent more than two weeks.

Part 2: deep dive into an algorithm:

  • Motiviation for Round 2

  • Proved main result about why it works carefully

  • Went through a working implementation.

  • Gives a sense for how algebraic number theory algorithms often actually work:

    • Theoretically nice construction, which doesn't lend itself to computation

    • Introduce new object that is maybe complicated, but can be efficiently computed.

    • Prove a result that makes that new object useful.

Now for the third:

Part 3: the whirlwind tour

  • We have 14 meetings left.

  • Here are 7 topics: I'm going to tell you a little bit about them all.

    1. Ideal class group (and unit group) of number fields

    2. Exactly linear algebra

    3. Finitely generated abelian groups

    4. Lattices (LLL)

    5. Elliptic curves

    6. Modular forms

    7. p-adic numbers

︠a3ed3304-dd0b-4057-ae0f-2f2440cf861b︠
a =Mod(4, 2016)

a.sqrt(all=True)
[2, 110, 142, 250, 254, 362, 394, 502, 506, 614, 646, 754, 758, 866, 898, 1006, 1010, 1118, 1150, 1258, 1262, 1370, 1402, 1510, 1514, 1622, 1654, 1762, 1766, 1874, 1906, 2014]