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Given a quadratic form Q(x,y)=x^2+y^2, (1) then Q(x,y)Q(x^',y^')=Q(xx^'-yy^',x^'y+xy^'), (2) since (x^2+y^2)(x^('2)+y^('2)) = (xx^'-yy^')^2+(xy^'+x^'y)^2 (3) = ...

The number of elements of a group in a given conjugacy class.

The order of the polynomial defining an algebraic curve.

z^p-y^p=(z-y)(z-zetay)...(z-zeta^(p-1)y), where zeta=e^(2pii/p) (a de Moivre number) and p is a prime.

A number of the form a_0+a_1zeta+...+a_(p-1)zeta^(p-1), where zeta=e^(2pii/p) is a de Moivre number and p is a prime number. Unique factorizations of cyclotomic integers fail ...

For a finite group of h elements with an n_ith dimensional ith irreducible representation, sum_(i)n_i^2=h.

A polynomial form which is the sum of two polynomial forms involving separate sets of variables.

The order of a finite field is the number of elements it contains.

The series sum_(k=1)^inftykr^k=sum_(k=1)^inftysum_(i=k)^inftyr^i=r/((1-r)^2), valid for 0<r<1.

If a Sylow 2-subgroup T of G lies in a unique maximal 2-local P of G, then P is a "strongly embedded" subgroup of G, and G is known.