This is notes for another section of Bump’s text on automorphic form.
When I was traveling to Taiyuan, I wrote this following Bump’s text on Automorphic forms and representations section 1.3 by filling out details and working on necessary exercises. I will continue doing this throughout the summer and next semester.
This is the notes based on the work by Tom Weston on Idelic Approach to Number Theory. Class_Number_through_Adeles_and_Ideles
- Introduced the concept of Algebraic Numbers and proved some of their properties: Countability (Cantor), finite degree field extension, algebraically closed in fields or rings
- Proved the existence of Transcendental Numbers (Liouville).
- Discussed some properties of minimal polynomials
- Unique monic minimal polynomial having a root as algebraic number
- Irreducible implies distinct roots in Complex field
- If a is algebraic, the ring of Q joined by a is equivalent to the field of Q joined by a
- Theorem: If K is a subset of L, and is a finite extension of field with characteristic 0, then L is a field extension of K.
- Introduced the concept of Algebraic Integer, and that any monic polynomial with coefficient being integer will be an algebraic integer
- Introduced the concept of Module, considered rings of algebraic integers as modules, connection with linear algebra.
- Tool: if d = 2,3 mod 4, then Z joined by root d will be normal, while d = 1 mod 4, Z joined by root d will be (1+root(d))/2.
- Euclidean Algorithm and it’s expansion into other domains, like polynomials and imaginary fields. So as the Bezout algorithm.
- Reviewed the Fundamental Theorem of Arithmetic.
- Theorem: Every prime number p = 1 mod 4 can be written as the sum of two squares. (Proof intuition, Wilson’s Theorem) More generalized Theorem: A natural number could be written as the sum of two squares iff its 3 (mod 4) prime factorizations have even power.
- Discussed why the Euclidean Algorithm fails in some field extensions.
- Solved equations in the third or higher power in other domains.