The famous number theory book Dedekind wrote and credited to his teacher
In 1863 Richard Dedekind published Lectures on Number Theory under Peter Gustav Lejeune Dirichlet's name and called it Dirichlet's book for the rest of his life. Yet he wrote nearly all of it himself, mostly after Dirichlet died, and its later supplements introduced the ideal, a concept modern algebra still runs on.
Algebraic number theory applies abstract algebra to whole numbers and fractions. Instead of attacking an equation head on, it asks structural questions: whether a system of numbers allows unique factorisation, how its ideals behave, what symmetries its fields have. Its roots go back to Diophantus of Alexandria in the third century, whose Arithmetica survives only in part, and further still to Babylonian lists of Pythagorean triples around 1800 BC.
Carl Friedrich Gauss turned a scatter of isolated results into a discipline. He wrote the Disquisitiones Arithmeticae at 21 and published it at 24 in 1801, pulling together work by Fermat, Euler, Lagrange and Legendre, fixing weak proofs and adding his own. His cryptic notes hinted at theories that others would develop decades later. Dirichlet followed with a class number formula in 1838 and 1839, which Jacobi praised as touching the utmost of human acumen, and he settled the cases of Fermat's Last Theorem for exponents 5 and 14.
Dedekind's ideals generalised the ideal numbers Ernst Kummer devised in 1843 while trying to prove Fermat's theorem. Fittingly, the word ring was coined later by David Hilbert, whose 1897 Zahlbericht unified the field, and the ideal was developed further above all by Emmy Noether. Ideals were needed because factorisation can go strange beyond ordinary integers. Among the Gaussian integers, 5 splits as (1 + 2i)(1 − 2i) and also as (2 + i)(2 − i), so primes must be treated as unique only up to units and order.
The twentieth century brought Emil Artin's reciprocity law and, around 1955, a startling guess by Goro Shimura and Yutaka Taniyama that elliptic curves and modular forms are secretly linked. Andrew Wiles proved enough of it to settle Fermat's Last Theorem. His June 1993 announcement contained a serious gap; the repaired proof, partly with Richard Taylor, appeared in September 1994.
Source: Algebraic number theory
