Beyond the Quartic Equation
One of the landmarks in the history of mathematics is the proof of the nonex- tence of algorithms based solely on radicals and elementary arithmetic operations (addition, subtraction, multiplication, and division) for solutions of general al- braic equations of degrees higher than four. This proof by the French mathema- cian Evariste Galois in the early nineteenth century used the then novel concept of the permutation symmetry of the roots of algebraic equations and led to the invention of group theory, an area of mathematics now nearly two centuries old that has had extensive applications in the physical sciences in recent decades. The radical-based algorithms for solutions of general algebraic equations of degrees 2 (quadratic equations), 3 (cubic equations), and 4 (quartic equations) have been well-known for a number of centuries. The quadratic equation algorithm uses a single square root, the cubic equation algorithm uses a square root inside a cube root, and the quartic equation algorithm combines the cubic and quadratic equation algorithms with no new features. The details of the formulas for these equations of degree d(d = 2,3,4) relate to the properties of the corresponding symmetric groups Sd which are isomorphic to the symmetries of the equilateral triangle for d = 3 and the regular tetrahedron for d — 4.
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algebraic equations alternating group axes Bring-Jerrard quintic Brioschi quintic 6.3-1 called Chapter coefficients complex numbers conjugacy class corresponding cube roots cubic equation defined derived E. R. Canfield edges equation algorithm equation of degree evaluation expressed field extension Figure finite following equation formulas Galois group Galois theory gives group A5 group Sn group theory groups of degree icosahedron improper rotation integral domain inverse irreducible isomorphic Kiepert algorithm linear Math mathematics metacyclic group method midpoints monic multiplication non-zero normal subgroup object obtained octahedron parameters pentagon permutation group poles polyhedral polynomials principal quintic 6.2-2 quadratic equation quartic quintic equation quotient rational regular polyhedra Riemann sphere rotation axis Section sextic equation soluble by radicals solution of algebraic solving splitting field square root subfield Substituting symmetric group symmetry operations symmetry point group tetrahedron theta functions transitive permutation groups transvectants trigonometric functions Tschirnhausen transformation values variable vertices Weierstrass elliptic functions zeros