1
Q
Euclidean domain
A
has a Euclidean valuation; a=qb+r with δ(r)<δ(b)
2
Q
principal ideal domain
A
every ideal is principal; I=aR
3
Q
unit
A
divides 1
4
Q
associated
A
equivalent up to unit
5
Q
irreducible
A
p=ab, then a or b is a unit
6
Q
prime
A
p|ab then p|a or p|b
7
Q
Unique Factorisation Domain
A
-every non-zero non-unit can be written as the product of finitely many irreducible elements
-unique up to order
8
Q
τ(n)
A
number of divisors
9
Q
σ(n)
A
sum of divisors
10
Q
multiplicative
A
f(1)=1 and for m,n coprime f(mn)=f(m)f(n)
11
Q
mobius function
A
square-free gives (-1)^s , else 0
12
Q
dirichlet product
A
Sum over divisors of n: f(n/d)g(d)
Number Theory
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