21. Many classic examples of this stem from algebraic number theory. 22. The algebraic numbers form a subfield of the real numbers. 23. In addition, they can also be closely approximated by algebraic numbers of degree 4, 24. It covers elementary and algebraic number theory, including Dirichlet's theorem. 25. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory. 26. His mathematical work was in algebraic number theory and abstract algebra, especially group theory. 27. All roots of polynomials with rational coefficients are algebraic numbers , by definition of the latter. 28. He introduced the Tamagawa numbers, which are measures for algebraic groups over algebraic number fields. 29. One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. 30. For a given algebraic number ? and algebraic number field " K ", the equation
