Introduction to Algebraic Integers — Extending the Notion of Integer
Using the Gaussian integers Z[i] and the Eisenstein integers Z[omega] as concrete examples, we introduce rings of algebraic integers. We demonstrate the failure of unique factorization in Z[sqrt(-5)] and show how ideals and prime ideal factorization restore uniqueness. We define the class number and discuss its significance.
1 The Gaussian Integers
Definition 1 (Gaussian integers).
The ring of Gaussian integers is , where .
Definition 2 (Norm).
The norm of a Gaussian integer is , where .
Theorem 3 (Multiplicativity of the norm).
.
Proof.
. □
Theorem 4 ( is a Euclidean domain).
is a Euclidean domain with respect to the norm . That is, for any with , there exist such that and .
Proof.
Write with . Choose integers with and , and set . Then satisfies . □
Corollary 5.
is a unique factorization domain (UFD).
Theorem 6 (Classification of Gaussian primes).
The primes (irreducible elements) of , up to multiplication by units , fall into three types:
(corresponding to the rational prime , since ).
where for a rational prime .
, where is a rational prime with .
2 The Eisenstein Integers
Definition 7 (Eisenstein integers).
Let . The ring of Eisenstein integers is , with norm .
Theorem 8.
is a Euclidean domain and hence a UFD.
Remark 9.
The Eisenstein integers play an essential role in the proof of Fermat's last theorem for the exponent (showing that has no nontrivial integer solutions).
3 The Failure of Unique Factorization
Theorem 10.
is not a UFD.
Proof.
Consider the norm on . We have
The norms are , , . There is no with (the equation has no integer solutions), and similarly none with . Therefore , , , and are all irreducible. Since and are not associates (their norms differ), the integer admits two essentially distinct irreducible factorizations. □
4 Ideals and Prime Ideal Factorization
Definition 11 (Ideal).
A nonempty subset of a ring is an ideal if for all and for all , .
Theorem 12 (Dedekind's theorem).
Every nonzero ideal of the ring of integers of an algebraic number field can be written uniquely (up to the order of the factors) as a product of prime ideals.
Remark 13.
Dedekind's theorem guarantees that even when unique factorization of elements fails, unique factorization of ideals still holds. In , although does not factor uniquely into irreducible elements, at the level of ideals we have , which is the unique prime ideal factorization.
Example 14.
In : where , and where .
5 The Class Number
Definition 15 (Ideal class group and class number).
Let denote the group of fractional ideals of and the subgroup of principal fractional ideals. The quotient group is called the ideal class group, and its order is the class number.
Theorem 16.
is a UFD if and only if .
Proof.
If , then every ideal is principal, making a principal ideal domain (PID). Every PID is a UFD (by a standard argument). Conversely, if is a UFD, then every ideal is principal (this follows from properties of irreducible elements), so . □
Example 17.
(so is a UFD), while (so is not a UFD). The imaginary quadratic fields () with class number are finite in number: they correspond to , a total of nine (by the theorem of Heegner–Stark–Baker).
Remark 18.
The class number is an important invariant that measures how far the ring of integers deviates from unique factorization. One can prove that the class number is always finite using the Minkowski bound, and concrete computations rely on the relationship between the discriminant of the field and the norms of ideals.
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