Algebra I

Syllabus

• Complex numbers as ordered pairs and algebraic objects; real and imaginary parts, arithmetic operations, conjugation and modulus.
• Geometric representation of complex numbers; polar form, argument, multiplication/division in trigonometric form; triangle inequality.
• Roots of unity, primitive n-th roots, extraction of n-th roots and their geometric interpretation.
• Polynomials over rings and fields: degree, leading coefficient, polynomial arithmetic and structure of polynomial rings.
• Polynomial functions, Horner scheme, factor theorem, multiplicity of roots and the Fundamental Theorem of Algebra (statement).
• Irreducibility over fields; Rational Root Test, Eisenstein criterion, Gauss lemma; unique factorization in polynomial rings (overview).
• Multivariate polynomials, symmetric polynomials, relations between roots and coefficients.
• Linear systems and Gaussian elimination; rank, existence and uniqueness of solutions; Cramer’s rule.
• Matrices and determinants: properties, computation, invertibility, Vandermonde determinant, determinant product rule.
• Basic abstract algebra concepts: groups, rings, fields; modular arithmetic and permutations.