-
(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,
- (Fraleigh 20.2) If
-
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 .
- Intuition: The set formed by
-
(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 .
- Proof: The set of all algebraic numbers is simply the Algebraic Closure of
- (Fraleigh 31.13) The set of all algebraic numbers forms a field
Topics
Links
- A First Course in Abstract Algebra 7th Edition by Fraleigh - some theorems in Number Theory can be studied using Abstract Algebra