Abstract Algebra Theory and Applications - Computer Science ...

270 CHAPTER 15 POLYNOMIALSProof. Suppose that p(x) generates a maximal ideal of F [x]. Then 〈p(x)〉 isalso a prime ideal of F [x]. Since a maximal ideal must be properly containedinside F [x], p(x) cannot be a constant polynomial. Let us assume that p(x)factors into two polynomials of lesser degree, say p(x) = f(x)g(x). Since〈p(x)〉 is a prime ideal one of these factors, say f(x), is in 〈p(x)〉 and thereforebe a multiple of p(x). But this would imply that 〈p(x)〉 ⊂ 〈f(x)〉, which isimpossible since 〈p(x)〉 is maximal.Conversely, suppose that p(x) is irreducible over F [x]. Let I be an idealin F [x] containing 〈p(x)〉. By Theorem 15.12, I is a principal ideal; hence,I = 〈f(x)〉 for some f(x) ∈ F [x]. Since p(x) ∈ I, it must be the casethat p(x) = f(x)g(x) for some g(x) ∈ F [x]. However, p(x) is irreducible;hence, either f(x) or g(x) is a constant polynomial. If f(x) is constant,then I = F [x] and we are done. If g(x) is constant, then f(x) is a constantmultiple of I and I = 〈p(x)〉. Thus, there are no proper ideals of F [x] thatproperly contain 〈p(x)〉.□Historical NoteThroughout history, the solution of polynomial equations has been a challengingproblem. The Babylonians knew how to solve the equation ax 2 + bx + c = 0.Omar Khayyam (1048–1131) devised methods of solving cubic equations throughthe use of geometric constructions and conic sections. The algebraic solution ofthe general cubic equation ax 3 + bx 2 + cx + d = 0 was not discovered until thesixteenth century. An Italian mathematician, Luca Paciola (ca. 1445–1509), wrotein Summa de Arithmetica that the solution of the cubic was impossible. This wastaken as a challenge by the rest of the mathematical community.Scipione del Ferro (1465–1526), of the University of Bologna, solved the “depressedcubic,”ax 3 + cx + d = 0.He kept his solution an absolute secret. This may seem surprising today, whenmathematicians are usually very eager to publish their results, but in the daysof the Italian Renaissance secrecy was customary. Academic appointments werenot easy to secure and depended on the ability to prevail in public contests. Suchchallenges could be issued at any time. Consequently, any major new discovery wasa valuable weapon in such a contest. If an opponent presented a list of problemsto be solved, del Ferro could in turn present a list of depressed cubics. He kept thesecret of his discovery throughout his life, passing it on only on his deathbed to hisstudent Antonio Fior (ca. 1506–?).Although Fior was not the equal of his teacher, he immediately issued a challengeto Niccolo Fontana (1499–1557). Fontana was known as Tartaglia (the Stammerer).As a youth he had suffered a blow from the sword of a French soldier duringan attack on his village. He survived the savage wound, but his speech was perma-