Galois Theory

Algebra began when quadratic equations were solved by al-Khowarizmi. Its next step was the solution of third and fourth degree equations, published by Cardano in [1545]. Equations of degree 5, however, resisted all efforts at similar solutions, until Abel [1824] and Galois [1830] proved that no such solution exists. Abel’s solution did not hold the germs of future progress, but Galois’s ideas initiated the theory that now bears his name, even though Galois himself lacked a clear definition of fields. The modern version has remained virtually unchanged since Artin’s lectures in the 1920s.