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[1] Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry).
For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2 , 5/4, and √ 2 are not. [8] The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers.
Number systems have progressed from the use of fingers and tally marks, perhaps more than 40,000 years ago, to the use of sets of glyphs able to represent any conceivable number efficiently. The earliest known unambiguous notations for numbers emerged in Mesopotamia about 5000 or 6000 years ago.
Number theory began with the manipulation of numbers, that is, natural numbers (), and later expanded to integers and rational numbers (). Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations. [15]
On the other hand, the maximal real subfields Q(cos(2π/2 n)) of the 2-power cyclotomic fields Q(ζ 2 n) (where n is a positive integer) are known to have class number 1 for n≤8, [8] and it is conjectured that they have class number 1 for all n. Weber showed that these fields have odd class number.
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