Notes on Leray's index theory

Notes on Firing Theory

Notes on Galois Theory

Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by AutE. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of AutE consisting of all automorphisms σ : E → E such that σ(a) = a for all a ∈ F . We write this last c...

Notes on Galois Theory

§1. Algebraic Extensions 2 §1.1. Field extensions 2 §1.2. Multiplicativity of degree 4 §1.3. Algebraic extensions 4 §1.4. Adjoining roots 5 §1.5. Splitting fields 5 §1.6. Algebraic closure 7 §1.7. Finite fields 9 §1.8. Composite field 9 §1.9. Exercises 10 §2. Galois Theory 11 §2.1. Separable extensions 11 §2.2. Normal extensions 13 §2.3. Main Theorem of Galois Theory 13 §2.4. Fields of invarian...

Notes on Probability Theory

I Axioms of Probability 2 I-A Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1a Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I-A1b Mutually Exclusive Sets or Disjoint Sets . . . . . . . . . . . . . . . . . . . 2...

Notes on Complexity Theory Notes Taken From

I assume that most students have encountered Turing machines before. (Students who have not may want to look at Sipser’s book [3].) A Turing machine is defined by an integer k ≥ 1, a finite set of states Q, an alphabet Γ, and a transition function δ : Q×Γk → Q×Γk−1×{L, S,R}k where: • k is the number of (infinite, one-dimensional) tapes used by the machine. In the general case we have k ≥ 3 and ...