prim := An integer p > 1 is called a prime number or
a prime if it has not positive divisors other than 1 and p. An integer n > 1 is called composite if it
is not prime. Euclid’s lemm := Let p be prime and a, b ∈ ℤ. Suppose p|ab then p|a or p|b. Fundamental Theorem of
Arithmeti := Every integer n > 1 can be expressed uniquely
(up to reordering) as a product of primes. Proof of the Fundamental
theorem of Arithmetic Case 1- if p|a, then we are done. Case
2- if p doesn’t divide a, then gcd(p, a) = 1
and ∃c ∈ ℤ : ab = pc
by the proposition. According to [[Bezouts_lemma]] ∃u, v ∈ ℤ, au + pv = 1 ⟹ (ba)u + p(bv) = b ⟹ p(cu) + p(bv) = b ⟹ p(cu + bv) = b
hence p is a multiple of b, (in other words p|b▫)
Algebrai := A number is algebraic if it is a root of a
non-zero polynomial with rational coefficients.
trancendenta := A number is trancendental if it is not
algebraic