Introduction to Volume 1, Fall 2013

18 . 782 Introduction to Arithmetic Geometry Fall 2013

Remark 4.2. For those familiar with category theory, one can define inverse limits for any category. In most cases the result will be another object of the same category (unique up to isomorphism), in which case the projection maps are then morphisms in that category. We will restrict our attention to the familiar categories of sets, groups, and rings. One can also generalize the index set {n} ...

Math 661 Fall 2013 Homework 1 Solutions

Proof. First suppose that a ∼H b, so that a = bk for some k ∈ H, and let ah (with h ∈ H) be an artibrary element of aH. Then we have ah = bkh = b(kh) ∈ bH, hence aH ⊆ bH. The proof of bH ⊆ aH is similar. Conversely, suppose that aH = bH. Since a ∈ aH = bH we have a = bk for some k ∈ H. Then a−1b = k−1 ∈ H, hence a ∼H b. (d) Prove that the map g 7→ ag is a bijection from H to aH. Proof. Consider...

Algebra Prelim Fall 2013

First we will prove a small lemma. Lemma 1. Let P be a p-Sylow subgroup of G. Let N be a normal subgroup of G such that P ⊂ N . Then all p-Sylow subgroups are in N . Proof. Let P̃ be a p-Sylow subgroup that is not P . Then by the Sylow Theorems, we know that gPg−1 = P̃ for some g ∈ G. Since P ⊂ N , and N is normal, we know that gPg−1 ⊂ N and hence P̃ ⊂ N . Thus all p-Sylow subgroups are in N . Now...

Honors A: Fall 2013

Proof. We prove the statement by induction on n. n = 1 case can be easily verified. Let’s assume the theorem is true for n − 1. We take an orthonormal basis [e1, · · · , en] for W and write Lei = ∑n j=1 ajiej. Since L is symmetric, we have aji = aij or A = [aij] is a symmetric matrix. From the lemma, any root of the characteristic polynomial of A is real. Take λ1 to be a root, then A−λ1I is sin...

Algebra Fall 2013