(number-theory-to-be-sorted)=
# To Be Sorted

## Definitions

:::{prf:definition} 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 $P(z)=0$.
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:::{prf:definition} Transcendental Numbers

A **transcendental number** is a complex number that is not algebraic.
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:::{prf:definition} 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 $P(z)=0$.
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:::{prf:definition} 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 $s_i\in S$.
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## Propositions

:::{prf:proposition}

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.
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:::{prf: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 $w=1/\pi$.
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## Theorems

:::{prf:theorem} 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.
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:::{prf:proof}

TODO
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:::{prf:theorem} Lindemann–Weierstrass theorem

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