Contents

• A. Fields

• B. Vector Spaces

• C. Homogeneous Linear Equations

• D. Dependence and Independence of Vectors

• E. Non-Homogeneous Linear Equations

• F. Determinants

Contents

• A. Extension Fields

• B. Polynomials

• C. Algebraic Elements

• D. Splitting Fields

• E. Unique Decomposition of Polynomials into Irreducible Factors

• F. Group Characters

• G. Applications and Examples to Theorem 13

• H. Normal Extensions

• I. Finite Fields

• J. Roots of Unity

• K. Noether Equations

• L. Kummer's Fields

• M. Simple Extensions

• N. Existence of a Normal Basis

• O. Theorem of Natural Rationality

Contents

• A. Solvable Groups

• B. Permutation Groups

• C. Solution of Equations by Radicals

• D. The General Equation of Degree n

• E. Solvable Equations of Prime Degree

• F. Ruler and Compass Constructions