How do you check if a number is an algebraic integer?
How do you check if a number is an algebraic integer?
- α ∈ K is an algebraic integer if there exists a monic polynomial f(x) ∈
- α ∈ K is an algebraic integer if the minimal monic polynomial of α over is in.
- α ∈ K is an algebraic integer if [α] is a finitely generated.
- α ∈ K is an algebraic integer if there exists a non-zero finitely generated -submodule M ⊂
How do you know if a number is algebraic?
Algebraic numbers include all of the natural numbers, all rational numbers, some irrational numbers, and complex numbers of the form pi + q, where p and q are rational, and i is the square root of −1. For example, i is a root of the polynomial x2 + 1 = 0.
What makes a number algebraic?
To be algebraic, a number must be a root of a non-zero polynomial equation with rational coefficients.
What is an example of algebraic number?
An algebraic number is any number that is the solution to a polynomial with rational coefficients. For example, 5 is an algebraic number because it is the solution to x - 5 = 0. The square root of 5 is also an algebraic number because it is the solution to x^2 - 5 = 0.
Is an algebraic integer?
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers.
Is zero an algebraic number?
Zero is algebraic, being a root of the polynomial (for instance). Every real or complex number is either algebraic or transcendental because the definition of a transcendental number is a number that is not algebraic. Transcendental just means "not algebraic".
Is Pi an algebraic number?
Therefore π is not algebraic, which means that it is transcendental.