Algebra

Algebra studies structures defined by operations and the laws, morphisms, and symmetries that organize them.

Over two fifths of the subject's articles are in polynomials and power series. Its largest topic, deep power-sum congruences, shows that for monic polynomials over a torsion-free \(\mathbb{Z}_{(p)}\)-algebra, congruences of the first \(N\) elementary symmetric functions modulo a divided-power ideal \(\mathfrak{a}\) are equivalent to congruences of the \(n\)-th power sums modulo \(n\mathfrak{a}\) for \(n \le N\). Orthogonal polynomials include Favard's theorem on three-term recurrences, real zeros of polynomials are counted by Sturm's theorem, Lie algebras have Ado's theorem and, in characteristic zero, the Levi decomposition, and associative algebras include Frobenius's classification of the finite-dimensional real division algebras and Jacobson's commutativity theorem.

• Associative rings and algebras
• Lie algebras and nonassociative structures
• Algebraic identities and operations
• Polynomials and power series
• Zeros of polynomials
• Orthogonal polynomials and polynomial sequences

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