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Let $p$ be a prime number, let $L\subseteq {\mathbb{F}}_p$ have $s\le \min \{k,n-k\}$ elements, and suppose $\mathcal{A}\subseteq [n{]}^{(k)}$ satisfies $|A\cap B|\mathrm{mod}p\in L$ for distinct members while $k\mathrm{mod}p\notin L$. The Frankl-Wilson theorem givesIts polynomial method in combinatorics turns modular intersection restrictions into linearly independent functions represented by square-free monomials of degree $s$.

For a prime number $p$, a nonprincipal ultrafilter on $\mathbb{N}$ givesThe Łoś theorem makes $K$ a field of characteristic $p$. For every natural number $m$, almost all factors have at least $m$ elements, so $K$ is infinite.

For a set $\pi$ of prime numbers, the integer localizationconsists of fractions whose denominator has prime factors only in $\pi$. As an additive group it is divisible by every prime in $\pi$.

Every cyclotomic polynomial ${\Phi }_n$ is an irreducible polynomial over $\mathbb{Q}$. If a monic irreducible factor $f\in \mathbb{Z}[X]$ contains a primitive root $\zeta$, then it also contains ${\zeta }^p$ for every prime number $p\nmid n$: otherwise, writing ${\Phi }_n=fg$, reduction modulo $p$ and the Frobenius endomorphism give $\bar{f}\mid \bar{g}(X{)}^p$. Some irreducible factor would then divide both $\bar{f}$ and $\bar{g}$, giving a repeated factor of $X^n-1$, contrary to separability of roots of unity. Iterating over the prime factors of every integer coprime to $n$ shows that $f$ contains all primitive $n$th roots, so $f={\Phi }_n$.

For a finite group $G$ and prime number $p$, the notationdenotes the irreducible characters whose irreducible character degree is coprime to $p$.

The largest prime factor $P^{+}(n)$ of an integer $n>1$ is the greatest prime number dividing $n$.

For a prime number $p$, write a positive integer as $x=p^{v_p(x)}u$ with $p\nmid u$. Its last nonzero base-$p$ digit is $u\mathrm{mod}p$, an element of $\{1,\dots ,p-1\}$. Assigning this digit as the color gives the last nonzero digit coloring used in the necessity proof of Rado's theorem.

For a prime number $p$, the binomial coefficient modulo $p$ factors into the corresponding digit binomial coefficients in the base-p expansions of the nonnegative integers $n,k$. Pad both expansions to the same length and use $\left(\genfrac{}{}{0ex}{}{a}{b}\right)=0$ when $b>a$. The identity follows by factoring $(1+x{)}^n$ into digit powers and using the Frobenius endomorphism; digit uniqueness identifies each coefficient.

Let $p$ be a prime number, $0\le a<p$, $1\le s<p$, and $n\ge a+s$. A multilinear polynomial over ${\mathbb{F}}_p$ of degree less than $s$ that vanishes on every Boolean vertex whose weight is not congruent to $a$ modulo $p$ vanishes everywhere. Alternating sums over intervals of $s$ free coordinates eliminate the possible nonzero weight levels one at a time. This proves independence of the auxiliary functions used for the uniform Frankl-Wilson theorem bound.

In the polynomial method in combinatorics, a set family of sets with common size residue $r$ modulo a prime number $p$ annihilates the displayed auxiliary functions. After multilinear reduction on the Boolean cube, functions with $|S|\le s-1$ have degree at most $s$. If $r$ avoids every $j=0,\dots ,s-1$ modulo $p$, these functions are linearly independent: evaluation at a set $T$ gives $(|T|-r){\sum }_{S\subseteq T}{c}_S$, and increasing $|T|$ eliminates coefficients. Combining them with diagonal intersection polynomials removes the lower-degree monomial count and strengthens the dimension bound from ${\sum }_{j\le s}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ to $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$.

A monoid automorphism is a bijective operation- and identity-preserving self-map of a monoid. Its inverse is also a monoid homomorphism. Permuting prime numbers and extending by prime factorization gives all automorphisms of the multiplicative monoid of positive natural numbers; extending by $0\mapsto 0$ gives automorphisms of the multiplicative monoid including zero.

Let $p>q$ be prime numbers. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv 1(\mathrm{mod}q)$, so $q\mid p-1$.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

Let $p$ be a prime number and $L\subseteq {\mathbb{F}}_p$ have $s$ elements. A set family whose member sizes modulo $p$ avoid $L$, and whose distinct pairwise intersection sizes modulo $p$ belong to $L$, has at most ${\sum }_{j=0}^{\min (s,n)}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ members. The intersection polynomials ${\prod }_{\ell \in L}({\sum }_{i\in A}{x}_i-\ell )$ have a diagonal, nonzero evaluation matrix on the family's characteristic vectors of sets. Multilinear reduction on the Boolean cube places these linearly independent functions in the space spanned by square-free monomials of degree at most $s$.

An odd prime is a prime number that is an odd integer. The only even number that is a prime number is $2$, since every other positive even number has $2$ as a proper divisor. Thus every prime number greater than $2$ is an odd prime. There are infinitely many odd primes, by Euclid's theorem.

For a prime number $p$, the ring of p-adic integers is the inverse limitEvery p-adic integer has a unique convergent expansion ${\sum }_{j\ge 0}{b}_j{p}^j$ with $0\le b_j<p$. It is a complete discrete valuation ring with uniformizer $p$ and residue field ${\mathbb{F}}_p$.