O'Reilly logo

Auxiliary Polynomials in Number Theory by David Masser

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

11

Transcendence I – Mahler

In Chapters 2 and 3 we proved the irrationality of certain numbers ξ. The results of Chapters 4 and 5 can be considered as modest extensions. For example the finiteness of the number of (x, y) in Z2 with x(x2−2y2) = y is equivalent to the fact that for all large η in Z, the number ξ with ξ(ξ2 − 2η2) = η is of degree 3 over Q; or the irreducibility of X4 − 2η3Xη gives a number of degree 4 over Q.

In the present chapter we wish to go further and prove that certain ξ are transcendental over Q.

In Chapters 2 and 3 the irrationality of ξ was proved by first assuming ξ in Q, then by some method constructing a sequence of small rational numbers ξn ≠ 0 (n = 0, 1, 2, . . .), multiplying by suitable denominators to get into ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required