Number Theory

• Nicomachus’ Theorem.

• (Fraleigh 20.1) Fermat’s Little Theorem. If and is a prime not dividing , then

  For all

  • (Fraleigh 20.2) If then for any prime
  • Intuition: The set is equivalent to when . Two facts are key here: First, none of the multiples of in the set are (because ) and second, each multiple is distinct .
  • Intuition The set forms a group under multiplication modulo . Thus from Lagrange’s Theorem, divides . In other words,

• Euler’s Phi Function is a function defined as

• (Fraleigh 20.8) Euler’s Theorem. If is an integer relatively prime to . Then

  • Intuition: The set formed by is finite. In fact, it forms a multiplicative group of order .

• (Fraleigh 20.10, Fraleigh 20.11) Let and be relatively prime to . For each the equation has a unique solution in .

  Another way to interpret this

  If , then for any , the congruence

  has as solutions all integers in precisely one residue class modulo ,

• (Fraleigh 20.12, Fraleigh 20.13) Let and let . Let . The equation has a solution in if and only if .

  The equation has exactly solutions in .

  Another way to interpret this

  Let . The congruence has a solution if and only if . When this is the case, the solutions are the integers in exactly distinct residue classes modulo .

• An element of that is algebraic over is an algebraic number. A transcendental number is an element of that is transcendental over

  • (Fraleigh 31.13) The set of all algebraic numbers forms a field
    • Proof: The set of all algebraic numbers is simply the Algebraic Closure of in .

Topics §

• Number Theory — Notation Guide

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh - some theorems in Number Theory can be studied using Abstract Algebra