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Syllabus designed by:
A thorough understanding of all concepts of high school mathematics.
- P. M. Cohn: Classic algebra. Wiley, 2000
- Complex numbers: real and imaginary part, operations, no zero divisors. Conjugate, absolute value (modulus). The complex numbers form a field. Quadratic equations with complex coefficients.
- The complex plane, argument, multiplication and division in polar form. The triangle inequality.
- The polar form of the n-th roots. The order of a complex number, primitive n-th roots of unity.
- Polynomials: equality, operations, leading coefficient, monic polynomial, degree.
- Polynomial functions. Horner scheme, the maximum number of zeroes. The Fundamental Theorem of Algebra (without proof). Multiple roots. The Rational Roots Test. Lagrange interpolation.
- Multivariate polynomials. Relationship between roots and coefficients. Symmetric polynomials, the Newton-formulas.
- Factorizations of polynomials: divisors, units, irreducible, prime polynomials, greatest common divisor. Long divison, uniqueness. Euclid’s algorithm for polynomials, relationship between irreducible and prime elements. The ring of polynomials over a field has unique factorization.
- Irreducible polynomials over the field of complex and real numbers, conjugate roots. Eisenstein's criterion. Cyclotomic polynomials, they are irreducible over the rationals. Primitive polynomials, the Lemmas of Gauss, unique factorization for polynomials with integer coefficients.
- Cubic equations, Cardano’s formula, Casus irreducibilis. Equations of degree four.
- Vectors and matrices: systems of linear equations, Gaussian elimination. Homogeneous systems of equations, the existence of a nontrivial solution. Cramer’s rule.
- Column vectors over a field, addition, multiplication by a scalar. Matrix multiplication and addition. The unit matrix, inverse.
- The determinant: definition, basic propeties, computing the determinant by elimination. Transpose, Vandermonde determinant. The determinant of the product of two matrices. Minors, cofactors, expansion, formula for the inverse matrix.
- Basic concepts in abstract algebra: commutative ring with identity, zero divisors, field. Multiplication by an integer. A field does not have zero divisors. Addition and multiplication modulo an integer m. The basic properties of binomial coefficients, the binomial theorem.
- Permutations, composition, inverse, their signs. The sign of the composition of two permutations. The number of even permutations.