Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendence
The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial P {\displaystyle P} with rational coefficients there will be a complex number α {\displaystyle \alpha } such that P ( α ) = 0 {\displaystyle P(\alpha )=0} . Transcendence theory is concerned with the converse question: given a complex number α {\displaystyle \alpha } , is there a polynomial P {\displaystyle P} with rational coefficients such that P ( α ) = 0 ? {\displaystyle P(\alpha )=0?} If no such polynomial exists then the number is called transcendental. More generally the theory deals with algebraic independence of numbers. A set of numbers {α1, α2, …, αn} is called algebraically independent over a field K if there is no non-zero polynomial P in n variables with coefficients in K such that P(α1, α2, …, αn) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where n = 1 and the field K is the field of rational numbers. A related notion is whether there is a closed-form expression for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.
History
Approximation by rational numbers: Liouville to Roth Use of the term transcendental to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function. The question of whether certain classes of numbers could be transcendental dates back to 1748 when Euler asserted that the number logab was not algebraic for rational numbers a and b provided b is not of the form b = ac for some rational c. Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim Joseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure. His original papers on the matter in the 1840s sketched out arguments using simple continued fractions to construct transcendental numbers. Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental. This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number e is transcendental. But his work did provide a larger class of transcendental numbers, now known as Liouville numbers in his honour. Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers. So if a number can be very well approximated by rational numbers then it must be transcendental. The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent. He showed that if α is an algebraic number of degree d ≥ 2 and ε is any number greater than zero, then the expression
| α − p q | < 1 q d + ε {\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{d+\varepsilon }}}}
can be satisfied by only finitely many rational numbers p/q. Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutions p/q for every d ≥ 2. In the twentieth century work by Axel Thue, Carl Siegel, and Klaus Roth reduced the exponent in Liouville's work from d + ε to d/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as the Thue–Siegel–Roth theorem, is ostensibly the best possible, since if the exponent 2 + ε is replaced by just 2 then the result is no longer true. However, Serge Lang conjectured an improvement of Roth's result; in particular he conjectured that q2+ε in the denominator of the right-hand side could be reduced to q 2 ( log q ) 1 + ϵ {\displaystyle q^{2}(\log q)^{1+\epsilon }} . Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence of many more numbers, such as the Champernowne constant. The theorem is still not strong enough to detect all transcendental numbers, though, and many famous constants including e and π either are not or are not known to be very well approximable in the above sense.
… excerpt ends here. Continue reading the full article.
