# Rings of Algebraic Integers

Fix an algebraic closure of . For example, could be the subfield of the complex numbers generated by all roots in of all polynomials with coefficients in .

Much of this course is about algebraic integers.

Definition 5.1.1 (Algebraic Integer)   An element is an if it is a root of some monic polynomial with coefficients in .

Definition 5.1.2 (Minimal Polynomial)   The of is the monic polynomial of least positive degree such that .

The minimal polynomial of divides any polynomial  such that , for the following reason. If , use the division algorithm to write , where . We have , so is a root of . However,  is the polynomial of least positive degree with root , so .

Lemma 5.1.3   If  is an algebraic integer, then the minimal polynomial of  has coefficients in  .

Proof. Suppose is the minimal polynomial of and is a monic integral polynomial such that . As mentioned after the definition of minimal polynomial, we have , for some . If , then some prime  divides the denominator of some coefficient of . Let be the largest power of  that divides some denominator of some coefficient , and likewise let be the largest power of  that divides some denominator of a coefficient of . Then , and if we reduce both sides modulo , then the left hand side is 0 but the right hand side is a product of two nonzero polynomials in , hence nonzero, a contradiction.

Proposition 5.1.4   An element is integral if and only if is finitely generated as a -module.

Proof. Suppose  is integral and let be the monic minimal polynomial of  (that is Lemma 5.1.3). Then is generated by , where  is the degree of . Conversely, suppose is such that is finitely generated, say by elements . Let  be any integer bigger than the degree of any . Then there exist integers such that , hence  satisfies the monic polynomial , so  is integral.

The rational number is not integral. Note that is not a finitely generated -module, since is infinite and .

Proposition 5.1.5   The set of all algebraic integers is a ring, i.e., the sum and product of two algebraic integers is again an algebraic integer.

Proof. Suppose , and let be the degrees of the minimal polynomials of , respectively. Then span and span as -module. Thus the elements for span . Since is a submodule of the finitely-generated module , it is finitely generated, so is integral. Likewise, is a submodule of , so it is also finitely generated and is integral.

Recall that a is a subfield  of such that the degree is finite.

Definition 5.1.6 (Ring of Integers)   The of a number field  is the ring

$x$ is an algebraic integer

The field of rational numbers is a number field of degree , and the ring of integers of is . The field of Gaussian integers has degree and . The field has ring of integers . Note that the Golden ratio satisfies . According to , the ring of integers of is , where .

Definition 5.1.7 (Order)   An in is any subring of such that the quotient of abelian groups is finite. (Note that must contain because it is a ring, and for us every ring has a .)

As noted above, is the ring of integers of . For every nonzero integer , the subring of is an order. The subring of is not an order, because does not have finite index in . Also the subgroup of is not an order because it is not a ring.

We will frequently consider orders in practice because they are often much easier to write down explicitly than . For example, if and is an algebraic integer, then is an order in , but frequently .

Lemma 5.1.8   Let be the ring of integers of a number field. Then and .

Proof. Suppose with in lowest terms and . The monic minimal polynomial of is , so if then Lemma 5.1.3 implies that is not an algebraic integer, a contradiction.

To prove that , suppose , and let be the minimal monic polynomial of . For any positive integer , the minimal monic polynomial of is , i.e., the polynomial obtained from by multiplying the coefficient of by , multiplying the coefficient of by , multiplying the coefficient of by , etc. If  is the least common multiple of the denominators of the coefficients of , then the minimal monic polynomial of has integer coefficients, so is integral and . This proves that .

In the next two sections we will develop some basic properties of norms and traces, and deduce further properties of rings of integers.

William Stein 2004-05-06