Part I

Foundations of Algebra

[504/79%]

Chapter ISets and maps[62/82%]

Section 1Operations on subsets. Calculation of the number of elements[6/50%]

Section 2Calculation of the number of maps and of the number of subsets. Binomial coefficients[13/69%]

Section 3Permutations[23/87%]

Section 4Recurrence relations. Induction[14/93%]

Section 5Summations[6/100%]

Chapter IIArithmetic spaces and linear equations[63/67%]

Section 6Arithmetic spaces[18/61%]

Section 7Rank of a matrix[19/53%]

Section 8Systems of linear equations[26/81%]

Chapter IIIDeterminants[52/88%]

Section 9Determinants of sizes two and three[2/100%]

Section 10Expanding a determinant. Inductive definition[7/71%]

Section 11Basic properties of a determinant[10/80%]

Section 12Expanding a determinant according to the elements of a row or a column[4/100%]

Section 13Calculating a determinant with the help of elementary operations[3/100%]

Section 14Calculating special determinants[1/100%]

Section 15Determinant of a product of matrices[4/100%]

Section 16Additional exercises[21/90%]

Chapter IVMatrices[81/72%]

Section 17Operations on matrices[29/79%]

Section 18Matrix equations. Inverse matrix[24/83%]

Section 19Special matrices[28/54%]

Chapter VComplex numbers[85/69%]

Section 20Complex numbers in algebraic form[11/73%]

Section 21Complex numbers in trigonometric form[14/71%]

Section 22Roots of complex numbers. Cyclotomic polynomials[27/67%]

Section 23Calculation of sums and products with the help of complex numbers[7/86%]

Section 24Complex numbers and geometry of a plane[26/65%]

Chapter VIPolynomials[161/87%]

Section 25Division with a remainder. Euclidean algorithm[9/89%]

Section 26Simple and multiple roots over fields of characteristic zero[14/93%]

Section 27Prime decomposition over

\mathbb{R}

\mathbb{C}

[14/100%]

Section 28Polynomials over the field of rationals and over finite fields[34/94%]

Section 29Rational fractions[5/80%]

Section 30Interpolation[14/57%]

Section 31Symmetric polynomials. Vieta formulas[34/76%]

Section 32Resultant and discriminant[17/100%]

Section 33Isolation of roots[20/90%]