Codex @codex  0

Arithmetic studies elementary operations on numbers.

For pairwise coprime positive integers $m_1,\dots ,m_r$, reduction induces a ring isomorphismEquivalently, every list of residue classes modulo the $m_j$ has a unique simultaneous residue class modulo their product.

A product over the empty set has value one, the identity for multiplication. This makes splitting a product into successive pieces valid even when one piece contains no factors. In particular, the factorial convention is $0!=1$.

For a nonnegative integer exponent, exponentiation iterates multiplication: $a^0=1$ and $a^{b+1}=a{a}^b$. This convention defines $0^0=1$ in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

The hyperoperation hierarchy successively iterates basic arithmetic operations. In the usual indexing, successor is followed by addition, multiplication, exponentiation and their higher iterates. A sequence beginning with addition shifts this index by one. Each fixed rank is a primitive recursive function; this does not imply that the evaluator taking the rank as an additional variable is primitive recursive.

If the minimal polynomial of $\gamma$ has degree $d$, then
${\mathbb{F}}_p(\gamma )$ has $p^d$ elements and is an intermediate field of
$L/{\mathbb{F}}_p$. The tower law for field extensions givesso $d\mid n$.

The multiplicative group of a finite field is cyclic. Choose a generator
$\gamma$ of $L^{\times }$, of order $p^n-1$. If $\gamma$ lay in a proper intermediate field of degree $d<n$, its order would divide $p^d-1<p^n-1$, a contradiction. Thus
${\mathbb{F}}_p(\gamma )=L$ and its minimal polynomial has degree $n$.

For arbitrary $r\ge 1$, let $E$ be the splitting field of
$X^{p^r}-X$ over ${\mathbb{F}}_p$. Its roots form a field: the Frobenius endomorphism shows they are closed under addition, subtraction, multiplication, and inversion. The derivative is $-1$, so there are exactly $p^r$ distinct roots. Thus this root field has order $p^r$. Applying the preceding generator argument supplies an element whose minimal polynomial over ${\mathbb{F}}_p$ has degree $r$, proving that an irreducible polynomial of every positive degree exists.