= Algebraic number theory
{wiki}
Algebraic number theory studies <integer factorization>, <divisibility>, <prime ideals>, and <units> after extending the <rational numbers> to a <number field>. Its basic objects are a <number field> $K$ and its <ring of integers> $\mathcal O_K$: the counterparts of $\mathbb Q$ and $\mathbb Z$. The <field norm> turns multiplication in $K$ into multiplication of <rational numbers>, and takes <algebraic integers> to <integers>. These connections let <linear algebra> and <Euclidean geometry> answer arithmetic questions.https://www.jmilne.org/math/CourseNotes/ANT.pdf[J. S. Milne, Algebraic Number Theory, introduction and Chapter 2]{ref}
A first example is $K=\mathbb Q(\sqrt2)$ with <ring of integers> $\mathbb Z[\sqrt2]$. The <integer> $7$ acquires the factorization
$$
7=(3+\sqrt2)(3-\sqrt2).
$$
Each factor has <field norm> $7$. Neither is a <unit>, and each is an <irreducible element>: any factorization into two nonunits would factor the <prime number> $7$ into two <integers> of absolute value greater than one. Arithmetic in a larger <ring> thus reveals structure invisible in ordinary <integer factorization>.
There is also a complication: <unique factorization> of elements can fail, as in the <failure of unique factorization in the ring of integers generated by square root of minus five>. The general replacement is <prime ideal factorization>. The <ideal class group> measures the obstruction to making every <ideal> a <principal ideal>.
The historical motivation is visible in <Richard Dedekind>'s original 1877 exposition: he begins with ordinary <integer factorization>, passes to <Gaussian integers>, and then uses a quadratic example to explain why <ideals> are needed. His later section establishes <unique factorization of ideals in a number field>. This is a direct record of the development of the theory, rather than a claim that all its ideas originated in one year.https://www.numdam.org/item/BSMA_1877_2_1_1_69_0.pdf[Richard Dedekind, Sur la théorie des nombres entiers algébriques, 1877, Section II, §§5–10]{ref}https://www.numdam.org/item/BSMA_1877_2_1_1_207_1.pdf[Dedekind, continuation, 1877, §25]{ref}
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