Algebraic Numbers

Algebraic numbers are complex numbers that are roots of non-zero polynomials with rational coefficients.

Definition and Minimal Polynomials

A number $\alpha \in \mathbb{C}$ is algebraic if there exists a polynomial $P(x) \in \mathbb{Q}[x]$ such that $P(\alpha) = 0$. The unique monic polynomial of lowest degree that $\alpha$ satisfies is called its minimal polynomial.

def evaluate_rational_poly(coeffs, x):
  res = 0
  for i, c in enumerate(coeffs):
      res += c * (x**i)
  return res

# Check if i (sqrt(-1)) is algebraic
# Minimal poly: x^2 + 1
coeffs_i = [1, 0, 1]
import math
val = evaluate_rational_poly(coeffs_i, 1j)
print(f"p(i) = {val}")

# Check if sqrt(2) + sqrt(3) is algebraic
# It satisfies x^4 - 10x^2 + 1 = 0
alpha = math.sqrt(2) + math.sqrt(3)
coeffs_alpha = [1, 0, -10, 0, 1]
print(f"p(sqrt(2)+sqrt(3)) = {evaluate_rational_poly(coeffs_alpha, alpha)}")

Algebraic Integers

An algebraic integer is a root of a monic polynomial with integer coefficients. For example, $\sqrt{2}$ is an algebraic integer, but $1/2$ is not (it is an algebraic number, but its minimal polynomial is $2x - 1$, which is not monic).

Transcendental Numbers

Numbers that are not algebraic are called transcendental. Proving a number is transcendental is usually much harder. Famous examples include $\pi$ and $e$.

import math

# Liouville's constant is transcendental
liouville = sum(10**(-math.factorial(n)) for n in range(1, 10))
print(f"Liouville's constant approx: {liouville}")
print("Transcendental numbers are 'uncountably' more common than algebraic ones.")