Barry Mazur proved uniform boundedness for elliptic curves over the rationals. If and are integers, rationals, or real numbers, then implies or . The step from rationals to reals is a major extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. Univariate algebraic equations over the rationals (i.e., with rational coefficients) have a very long history. All three topologies coincide and turn the rationals into a topological field. As shown above, the indicator function on the rationals is not Riemann integrable. It is less than all positive rational numbers, and greater than all negative rationals. If "B" has a smallest element among the rationals, the cut corresponds to that rational. Taking logs shows that formula 52 are linearly dependent over the rationals. Likewise, despite much interest, the problem for equations over the rationals remains open. The order type of the rationals is usually denoted formula 6. It follows that it is linear over the rationals, thus linear by continuity. The rationals are the smallest field with characteristic zero. The basic ingredients are hash tables that map rationals to strings.