Quadratic Number Fields

Number Theory, Math 413, Spring 2003

1. Simple Examples
2. Basic Operations & Definitions
3. Conjugates & the Galois Group
4. Algebraic Integers
5. Units
6. Primes & Splitting
7. Unique Factorization

1. Computational Tools
2. Tables
3. Examples
   1. Q[√13]
   2. Q[√-5]
   3. Fifth Cyclotomic Field
4. Open Questions
5. References

Readings:

• [Rose96, Sect 5.1]

1. Simple Examples

Gaussian Integers

Eisenstein Integers

2. Basic Operations & Definitions

We first view a quadratic number field naively as the set of all polynomials in the square root of some non-square, for example Q[√5]. This representation allows us to define most of the operations we want, but for some analysis we will have to use a more subtle point of view.

Addition & Multiplication: We add and multiply two elements of the quadratic number field in the usual way:

• (x + y√d) + (z + w√d) = (x + z) + (y + w)√d
• (x + y√d) (z + w√d) = (xz+ yw(√d)²) + (xw + yz)√d = (xz+ ywd) + (xw + yz)√d

Division takes a bit more thought, but can be done in a manner analogous to what we did in the Gaussian Integers, using something like complex conjugation to rationalize the denominator.

Division: We divide and then rationalize the denominator:
(x+y√d)/(z+w√d)
= ((x+y√d)(z-w√d))/((z+w√d)(z-w√d))
= ((xz-ywd)+(yz-xw)√d)/(z²-w²d)
= ((xz-ywd)/(z²-w²d)+√d(yz-xw)/(z²-w²d))

In order to rationalize the denominator we multiply by something other than the complex conjugate (unless d<0), but a more general conjugate, where √d → -√d. We will generalize this still further in the next section.

3. Conjugates & the Galois Group

Defn: conjugate

Defn: trace

Defn: norm

Defn: minimal polynomial

4. Algebraic Integers

Defn: α is an algebraic integer if it is the root of a polynomial whose coefficients are all rational integers.

In order to reduce (or increase) confusion we refer to the usual integers, {...,-2,-1,0,1,2,...}=Z⊂Q, as the rational integers.

We may specify that the minimal polynomial, m_α(x) (when the number field is viewed as an extension of Q) rational integers for all of its coefficients.

Characterizing the algebraic integers in a number field is not an easy task in general, but in the quadratic number field case we can easily characterize the algebraic integers. Consider the quadratic number field Q[√d]. We ask if the general element α=(a+b√d)c is an algebraic integer.

α=(a+b√d)c, assuming WLOG that gcd(a,b,c)=1.
m_&alpha(x) = x² + Tr(α)x + N(α)
= x² + (2a/c)x + (a²-db²)/c²
So α is an integer iff both (2a/c) and (a²-db²)/c² are rational integers.
Claim: For no prime p≠2 does p divide c
Were this true, then p|c|a (as (2a/c) is a rational integer). Thus, as (a²-db²)/c² is a rational integer and d is squarefree, we have that p|b and gcd(a,b,c)≠1, a contradiction. Thus c is a power of 2. In fact, the same argument shows that c is either 1 or 2.
The case c=1 is easy, a and b can take any integer values.
The case c=2 is equivalent to requiring that (a²-db²)/4 is a rational integer. Thus (a²-db²)=0(mod 4). The case a²=b²=0 (mod 4) is excluded, as then 2 divides gcd(a,b,c). Thus a² = db² = 1 (mod 4). Thus requires that d = 1 (mod 4) and a = b (mod 2). So we get:

The algebraic integers in Q[√d] have the form:

• {(a + b√d)/2}, where a = b (mod 2) if d=1 (mod 4)
• {a + b√d} if d={0,2,3} (mod 4)

It is common to reformulate this in terms of an element ω. The algebraic integers in Q[√d] are the Z-linear combinations of {1, ω}, where:

• ω = (1 + √d)/2 if d = 1 (mod 4)
• ω = √d if d = {0,2,3} (mod 4)

Thm: The algebraic integers form a ring with closure under addition and multiplication.

proof: In the case of quadratic number fields this is easily proven by looking at the form algebraic integers take in this case (see above).

5. Units

Thm: (Dirichlet's Unit Theorem)

6. Primes & Splitting

7. Unique Factorization

A. Computational Tools

• Mathematica: While Mathematica is able to manipulate Gaussian Integers but has no built in tools for any other number fields.
In[1]:= FactorInteger[2, GaussianIntegers->True]
• Maple Commands
>

B. Tables

Examples

1. Q[√13]
   The real quadratic field Q(√13) has a ring of integers O(√13) = <1, ω>, where ω = (1+√13)/2.
   The group of units is U(√13) = {±η₀^k}, where η₀ = 1 + ω = (3+√13)/2.
   Q(√13) has unique factorization and a rational prime p splits iff p = {1, 3, 4, 9, 10, 12} (mod 13)
   Units: α = (a+b√13)/2 is a unit iff N(α) = (a²+13b²)/4 = ±1. Thus (a²+13b²) = ±4. This generalized Pell Equation can be solved using the methods from earlier section and the fundamental solution is a=3 and b=1. Thus the fundamental unit, generating the infinite part of the group of units is η₀ = (3+√13)/2
   We see that the integers in Q[√13] have unique factorization as ...
   The small primes split as:
   • p = 2 is irreducible (2 is inert)
   • p = 3 splits as 3 = (-1)(7-2√13)(7+2√13). Note that N(3)=9 and N(7±2√13)=3.
   • p = 5 is irreducible (5 is inert)
   • p = 7 is irreducible (7 is inert)
   • p = 11 is irreducible (11 is inert)
   • p = 13 splits as as 13 = (√13)² As one of the irreducible factors of 13 occurs to a power greater than one we say that 13 ramifies.
   • p = 17 splits as 17 = (15-4√13)(15+4√13). Note that N(17)=17² and N(15±4√13)=17.
2. Q[√-5]
3. Fifth Cyclotomic Field

D. Open Questions

1. Class Number: Is every natural number equal to the class number of some quadratic field with negative discriminant? positive discriminant? (equivalent to similar question for quadratic forms)
2. Real with Unique Factorization: Are there an infinite number of real quadratic fields with unique factorization. (Conjectured by Guass.)

E. References

[Cohn62] Advanced Number Theory, by H. Cohn, Dover, 1980 (originally published in 1962 by Wiley as A Second Course in Number Theory)

[ST01] Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed, by Stewart & Tall, Peters Publ, 2001