algebraic integer
Quick Summary
An algebraic integer is a number that is a solution to a polynomial equation with integer coefficients, where the leading coefficient (the coefficient of the highest power of the variable) is 1. This concept extends the idea of integers to a broader set of numbers that can be manipulated algebraically.12
For example, the square root of 2 is an algebraic integer because it is a solution to the equation (x^2 - 2 = 0). Numbers like (\sqrt{2}) can be handled arithmetically, similar to whole numbers, and mathematicians have even defined prime numbers within this system.2
The theory of algebraic integers was significantly developed by mathematicians like Carl Friedrich Gauss and Richard Dedekind. Dedekind used ideals, which are specific sets of algebraic integers, to create a more abstract and structural understanding of number theory. This work laid the foundation for modern algebra and abstract mathematical concepts.23
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mathematicsDevelopments in pure mathematics…two reasons: the theory of algebraic integers forms part of it, because algebraic integers naturally form into rings; and (as Kronecker and Hilbert had argued) algebraic geometry forms another part. The rings that arise there are rings of functions definable on the curve, surface, or manifold or are definable on…
Read More…two reasons: the theory of algebraic integers forms part of it, because algebraic integers naturally form into rings; and (as Kronecker and Hilbert had argued) algebraic geometry forms another part. The rings that arise there are rings of functions definable on the curve, surface, or manifold or are definable on…
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modern algebraRings in number theory…Leopold Kronecker used rings of algebraic integers. (An algebraic integer is a complex number satisfying an algebraic equation of the form xn + a1xn−1 + … + an = 0 where the coefficients a1, …, an are integers.) Their work introduced the important concept of an ideal
Read More…Leopold Kronecker used rings of algebraic integers. (An algebraic integer is a complex number satisfying an algebraic equation of the form xn + a1xn−1 + … + an = 0 where the coefficients a1, …, an are integers.) Their work introduced the important concept of an ideal
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