Algebra

In mathematics, algebra is the study of examining, manipulating, and solving equations, inequalities, and other mathematical expressions. Algebra revolves around the concept of the variable, an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with algebraic manipulation. If this is not what you are looking for, you might try the page on abstract algebra.

Study Guides to Algebra

• Introductory topics in algebra
• Intermediate topics in algebra
• Olympiad number topics in algebra

Intermediate Topics

• Equations
  • Systems of equations
• Factoring
  • Simon's Favorite Factoring Trick
• Functions
  • Exponentials
  • Greatest integer function
  • Graphing
  • Least integer function
  • Logarithms
  • Polynomials
  • Functional Equations
• Inequalities
  • Trivial inequality
  • Cauchy-Schwarz Inequality
  • Arithmetic Mean-Geometric Mean Inequality
  • Triangle Inequality
• Newton sums
• Trigonometry
  • DeMoivre's Theorem
  • Roots of unity
• Continued fractions

Olympiad Topics

• Functions
  • Greatest integer function/Least integer function
  • Polynomials
  • Functional Equations
• Inequalities
  • Arithmetic Mean-Geometric Mean Inequality
  • Cauchy-Schwarz Inequality
  • Chebyshev's Inequality
  • Geometric inequalities
  • Hölder's Inequality
  • Isoperimetric inequalities
  • Jensen's Inequality
  • Minkowski Inequality
  • Muirhead's Inequality
  • Power Mean Inequality
  • Rearrangement Inequality
  • Schur's Inequality
  • Smoothing
  • Triangle Inequality
  • Trigonometric inequalities
  • Trivial Inequality

More Advanced Topics in Algebra

• Group theory
• Ring theory
• Field theory
• Galois theory
• Homological algebra
• Lie theory
• Algebraic geometry