Equivalence of Finite and Integral Extensions Algebra Galois Group of Cyclotomic Extensions Over Q Algebra Linear Independence of Eigenvectors for Distinct Eigenvalues Linear Algebra Root String Theorem Algebra Three by Three Lemma Algebra Homotopy Uniqueness Theorem for Lifted Chain Maps Algebra Chambers and Positive Systems of a Root System Algebra Artin-Rees Lemma Algebra Algebra Area