To Be Sorted

Definitions

Definition 4 (Algebraic Numbers)

An algebraic number is a complex number that is a root of a non-zero polynomial equation with integer coefficients. i.e., \(z\in\mathbb{C}\) is algebraic if there exists a polynomial \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\) with \(a_i\in\mathbb{Z}\) such that \(P(z)=0\).

Definition 5 (Transcendental Numbers)

A transcendental number is a complex number that is not algebraic.

Definition 6 (Algebraic Integers)

An algebraic integer is a complex number that is a root of a monic polynomial equation with integer coefficients. i.e., \(z\in\mathbb{C}\) is algebraic integer if there exists a polynomial \(P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\) with \(a_i\in\mathbb{Z}\) such that \(P(z)=0\).

Definition 7 (Algebraic Independence)

A subset \(S\) of a field \(L\) is algebraically independent over a subfield \(K\) if there exists no non-zero polynomial \(P(x_1, x_2, \ldots, x_n)\) with coefficients in \(K\) such that \(P(s_1, s_2, \ldots, s_n)=0\) for \(s_i\in S\).

Propositions

Proposition 3

For all transcendental numbers \(z,w\), the following statements hold:

  1. \(1/z\) is transcendental.

  2. \(z+w\) is not necessarily transcendental.

  3. \(zw\) is not necessarily transcendental.

Proof. For 1., suppose for contradiction \(1/z\) is algebraic. Then there exists \(a_i\in\mathbb{Z}\) such that \(a_n(1/z)^n+a_{n-1}(1/z)^{n-1}+\cdots+a_1(1/z)+a_0=0\). Multiplying both sides by \(z^n\), we have \(a_n+a_{n-1}z+\cdots+a_1z^{n-1}+a_0z^n=0\). This implies that \(z\) is algebraic.

For 2., take \(z=\pi\) and \(w=-\pi\).

For 3., take \(z=\pi\) and \(w=1/\pi\).

Theorems

Theorem 4 (Gelfond-Schneider Theorem)

For all algebraic numbers \(a,b\) such that \(a\neq 0,1\) and \(b\) is not a rational number, \(a^b\) is transcendental.

Proof. TODO

Theorem 5 (Lindemann–Weierstrass theorem)