Units are the reversible elements of a ring of integers, and Dirichlet’s unit theorem says their apparent infinity has a finite-dimensional lattice structure.
In Z, the only units are ±1. In Z[√2], infinitely many units appear: 1+√2, 3+2√2, 7+5√2 and their inverses and signs. The theorem explains this change. After taking logarithms of absolute values under every archimedean embedding, multiplication becomes addition and the free part of the unit group becomes a lattice in a hyperplane.
This is Guide 19 in the Bukit Timah Tutor Computational Number Theory series. It builds on number-field arithmetic in Guide 16 and lattice ideas in Guide 15. Return to the BTT Mathematics Hub for the wider Mathematics estate.
1. Units in a number field
Let K be a number field with ring of integers O_K. A unit u is an algebraic integer with an inverse that is also an algebraic integer.
Equivalently, the principal ideal (u) equals O_K. In many simple cases, norm gives a quick test: if u is a unit then N_K/Q(u)=±1. Conversely, an algebraic integer of norm ±1 is a unit.
The unit group is written O_K^×. It is an abelian group under multiplication.
2. The first contrast: Q and Q(√2)
For K=Q, O_K=Z and the unit group is {±1}, finite.
For K=Q(√2), O_K=Z[√2]. The element 1+√2 has norm
(1+√2)(1−√2)=−1.
Therefore it is a unit, with inverse −(1−√2)=√2−1.
Powers of 1+√2 give infinitely many distinct units. The ring has moved from a finite unit group to an infinite one even though the field degree is only two.
3. Archimedean embeddings
Let K have degree n. Its embeddings into C consist of r1 real embeddings and 2r2 nonreal complex embeddings occurring in conjugate pairs, with
n = r1+2r2.
For Q(√2), the two embeddings are
σ1(a+b√2)=a+b√2, σ2(a+b√2)=a−b√2.
Thus r1=2,r2=0.
For an imaginary quadratic field, r1=0,r2=1.
4. Multiplication becomes addition after logarithms
For nonzero α, define logarithmic coordinates using absolute values of its embeddings. One standard convention is
L(α)=(log|σ1(α)|,…,log|σ_r1(α)|,
2log|τ1(α)|,…,2log|τ_r2(α)|),
where one representative τ_j is chosen from each complex-conjugate pair.
Because log|xy|=log|x|+log|y|, we have
L(αβ)=L(α)+L(β).
This turns the multiplicative unit problem into an additive lattice problem.
5. The product formula creates a hyperplane
The field norm is the product of all embeddings, with complex conjugate pairs contributing squared absolute values. Therefore
sum of coordinates of L(α) = log|N_K/Q(α)|.
If u is a unit, |N(u)|=1, so the sum is zero.
Thus logarithmic unit vectors lie in the hyperplane
x1+…+x_(r1+r2)=0.
This hyperplane has dimension r1+r2−1.
6. Dirichlet’s unit theorem
Dirichlet’s unit theorem states
O_K^× ≅ μ(K) × Z^(r1+r2−1),
where μ(K) is the finite group of roots of unity contained in K.
The free rank is therefore
r = r1+r2−1.
After the logarithmic embedding, the free part maps to a full lattice in the zero-sum hyperplane.
This is a structural theorem: it tells us how many independent infinite unit directions exist before we know their explicit generators.
7. Unit ranks in small fields
For Q, r1=1,r2=0, so the rank is zero. Only the finite roots of unity remain: ±1.
For a real quadratic field, r1=2,r2=0, so the rank is one. There is one fundamental infinite unit direction.
For an imaginary quadratic field, r1=0,r2=1, so the rank is zero. Its unit group is finite.
For a totally real cubic field, r1=3,r2=0, so the rank is two.
8. The Q(√2) logarithmic lattice
Take ε=1+√2. Its conjugate is 1−√2, whose absolute value is √2−1=1/(1+√2).
Therefore
L(ε)=(log(1+√2), −log(1+√2)).
Every power ε^k maps to
k·L(ε).
The infinite unit group has become a one-dimensional lattice on the line x+y=0.
The sign −1 maps to the zero vector because both absolute values are one. This is why roots of unity disappear under the logarithmic map: they form the torsion kernel.
9. Fundamental units and Pell equations
A real quadratic field Q(√d) has unit rank one. A fundamental unit ε is a generator of the free part up to sign and possibly the choice of norm ±1 convention.
For Q(√2), ε=1+√2 is a fundamental unit with norm −1. Its square is 3+2√2 with norm +1.
The norm equation
a²−2b²=±1
is exactly the Pell/negative-Pell unit equation in this field. The continued-fraction machinery from Guide 7 computes the fundamental unit efficiently in quadratic fields.
10. What the regulator measures
Choose independent units ε1,…,ε_r generating the free part up to finite index. Their logarithmic vectors form a lattice in the zero-sum hyperplane.
The regulator R_K is the covolume of the full unit lattice, expressed by the absolute determinant of an appropriate r×r logarithmic matrix after one dependent coordinate is removed. Conventions for complex scaling can shift the matrix representation, so a computational reference should state its convention.
For rank one in Q(√2), using the standard deleted-coordinate convention:
R_K = log(1+√2).
Numerically this is about 0.88137.
11. Regulator is not the size of one arbitrary unit
If we use ε² instead of ε as a generator candidate, the logarithmic step doubles and its one-dimensional covolume becomes 2R_K. That subgroup has index two in the full free unit lattice.
Therefore computing a set of independent units is not enough. One must know whether they generate the full unit group modulo torsion or a finite-index subgroup.
The regulator of the subgroup and the regulator of the full unit group differ by that index.
12. Independence becomes linear independence of logs
Suppose units u1,…,us satisfy
u1^a1 ··· us^as = root of unity.
Applying L gives
a1L(u1)+…+asL(us)=0.
Thus multiplicative independence modulo torsion becomes linear independence of logarithmic vectors.
This is algorithmically useful because approximate real linear algebra can suggest dependence, while exact algebraic verification can confirm a relation.
13. Approximate logs need exact certificates
Embeddings of high-degree algebraic numbers are usually evaluated numerically. A near-zero determinant of a log matrix may indicate dependence or may reflect insufficient precision.
A robust computation increases precision, tracks error bounds and verifies any discovered multiplicative relation exactly in the number field.
For a claimed unit u, verify u∈O_K and N(u)=±1, or explicitly compute an algebraic-integer inverse. For a claimed relation, multiply the exact units using exact field arithmetic.
14. Searching for units through geometry
Minkowski embeddings convert O_K into a lattice in Euclidean space. Small algebraic integers correspond to lattice points in bounded regions.
One broad computational strategy searches for algebraic integers with smooth or controlled principal ideals, extracts multiplicative relations and combines them to produce units.
In higher-degree fields, direct enumeration of all possible units by coefficient size is ineffective because units can have extremely large coordinates even when the logarithmic regulator is moderate.
15. Relation matrices and units
Suppose principal ideals factor over a chosen factor base of prime ideals:
(α_j)=p1^e1j ··· pm^emj.
Each factorisation gives an exponent column e_j. An integer relation among these exponent columns whose total ideal exponent is zero yields a quotient/product of the α_j whose principal ideal is O_K—hence a unit, after accounting for exact equality and torsion.
The integer kernel of the relation matrix is computed with Hermite or Smith normal forms. Guide 18 therefore supplies the exact linear algebra behind this unit-search route.
16. Units and the class group are coupled
Relation collection in algebraic number theory often computes class-group information and unit information together. Ideal relations determine the quotient of the factor-base lattice, while dependencies among principal relations produce units.
This is not accidental. Both problems ask what multiplicative ideal information disappears when passing to principal ideals.
Guide 20 develops the class-group side of the same relation matrix.
17. Roots of unity: the torsion part
The finite subgroup μ(K) must be determined separately from the free units. In Q(√2), μ(K)={±1}.
Imaginary quadratic fields can contain more roots of unity: Q(i) contains four, and Q(√−3) contains six. Most imaginary quadratic fields contain only ±1.
The logarithmic map sends every root of unity to zero, so logs alone cannot distinguish these torsion elements.
18. A simple regulator calculation
For ε=1+√2:
σ1(ε)=1+√2, |σ2(ε)|=√2−1=(1+√2)^−1.
Let λ=log(1+√2). Then L(ε)=(λ,−λ). Delete the second coordinate. The 1×1 logarithmic matrix is [λ], so its absolute determinant is λ.
Thus R=λ≈0.8813735870 under this convention.
For ε²=3+2√2, the same calculation yields 2λ. This is the regulator of the index-two subgroup generated by ε², not the full unit group.
19. Unit equations
Once fundamental units are known, every unit can be written
ζ ε1^k1 ··· ε_r^kr
with ζ a root of unity and integer exponents k_i.
Equations such as u+v=1 with u and v restricted to units or S-units become finite after deep height bounds are applied, but their complete solution is much more sophisticated than merely enumerating small exponent vectors.
The unit-group representation is the essential first step because it converts an infinite multiplicative search into an integer-exponent search.
20. Precision and conditioning
A regulator matrix can be ill-conditioned when log vectors are nearly dependent. Floating determinant calculation may then lose many digits.
Computational systems use extra precision, lattice reduction, interval or ball arithmetic, and exact relation checks. The displayed decimal regulator should be treated as an approximation to an exact logarithmic expression.
Publishing many decimal digits does not improve the proof unless the precision method justifies them.
21. Common mistakes
1. Forgetting the roots-of-unity torsion factor. 2. Using rank n−1 instead of r1+r2−1. 3. Omitting the factor of two in the complex-coordinate log convention without adjusting the regulator convention. 4. Treating one large unit as automatically fundamental.
5. Calling independent units a full system of fundamental units without an index check. 6. Trusting a near-zero floating determinant as exact dependence. 7. Confusing element norm with Euclidean length of the embedding vector. 8. Assuming an imaginary quadratic field has an infinite unit rank.
22. Practice set
1. Define a unit in O_K. 2. Prove 1+√2 is a unit. 3. Find its inverse. 4. State r1,r2 and the unit rank for Q(√2).
5. State the rank for an imaginary quadratic field. 6. Compute the logarithmic vector of 1+√2. 7. Why do the coordinates sum to zero? 8. What is the torsion subgroup of Q(√2)?
9. Compute the standard rank-one regulator. 10. Why does 3+2√2 generate an index-two subgroup of the free unit group? 11. How does multiplicative independence appear under logarithms? 12. Why must floating relations be checked exactly?
13. How can a relation matrix produce units? 14. Why are class-group and unit computations often coupled? 15. What information does the regulator measure? 16. Give one reason direct coefficient enumeration is poor in higher-degree fields.
23. Answers
1. An algebraic integer whose inverse is also an algebraic integer. 2. Its norm is −1. 3. √2−1. 4. r1=2,r2=0, rank one.
5. Zero. 6. (λ,−λ), λ=log(1+√2). 7. The absolute norm of a unit is one. 8. {±1}.
9. log(1+√2)≈0.8813735870. 10. It equals (1+√2)², so its exponent lattice is 2Z instead of Z. 11. Integer multiplicative relations become integer linear relations among log vectors. 12. Numerical cancellation can mimic an exact relation.
13. Integer dependencies that cancel all prime-ideal exponents leave principal generators whose quotient/product has trivial principal ideal and hence is a unit. 14. They use the same ideal relations; quotient information gives class groups while kernels give units. 15. The covolume of the logarithmic free-unit lattice under a stated convention. 16. Fundamental units may have very large algebraic coordinates.
Sources and further study
MIT 18.785 Number Theory I, Lecture 15: Dirichlet’s Unit Theorem, develops the theorem through the logarithmic/Minkowski viewpoint. Henri Cohen’s A Course in Computational Algebraic Number Theory treats unit and regulator algorithms computationally. SageMath’s number-field documentation provides implementation-oriented functions for units, embeddings and regulators.
Continue through Batch 05
Use Guide 18: Smith & Hermite Normal Forms for the relation-matrix machinery. Continue to Guide 20: Class Group Algorithms, Minkowski Bounds, Relation Matrices and Principal Ideal Tests.