Fact-checked by Grok 2 weeks ago

Multiplicative group

In abstract algebra, the multiplicative group of a ring R, denoted R^\times, is the set of units in R—the elements that admit multiplicative inverses within the ring—forming an abelian group under the ring's multiplication operation when R is commutative.^[1] This structure generalizes the familiar group of nonzero elements under multiplication in fields and plays a fundamental role in understanding the invertible elements of algebraic systems. For a field F, the multiplicative group F^\times consists precisely of all nonzero elements of F, as every such element has a multiplicative inverse by definition. In the case of finite fields with q elements, F^\times is a cyclic group of order q-1, generated by a primitive element, which has significant implications for applications in coding theory and cryptography.^[2] The structure of these groups often reveals deep properties of the underlying field, such as the existence of roots of unity or the solvability of polynomial equations. Prominent examples include the multiplicative group (\mathbb{Z}/n\mathbb{Z})^\times of integers modulo n, comprising the residue classes coprime to n under modular multiplication, with cardinality given by Euler's totient function \phi(n).^[3] This finite abelian group is isomorphic to a product of cyclic groups and is central to number-theoretic concepts like Fermat's Little Theorem and the structure of Euler's theorem.^[3] Another key example is the multiplicative group \mathbb{C}^\times of nonzero complex numbers, which contains the circle group S^1 = \{ z \in \mathbb{C} : |z| = 1 \} as a subgroup and admits polar decomposition that highlights its connection to rotation and scaling symmetries.^[4] These groups exemplify how multiplicative structures encode geometric and arithmetic information across diverse mathematical domains.

Fundamentals

Definition

In abstract algebra, a multiplicative group is a group (G, \cdot) equipped with a binary operation \cdot interpreted as multiplication, where the identity element is $1 and every element g \in G has a multiplicative inverse g^{-1} \in G such that g \cdot g^{-1} = g^{-1} \cdot g = 1.^[5] The operation is associative, typically inherited from the surrounding algebraic structure, such as a ring or field, and the set G must be closed under this multiplication.^[6] Such groups commonly arise as the set of units in a ring R, denoted R^\times or U(R), consisting of all elements u \in R that possess multiplicative inverses within R; this set forms a group under the ring's multiplication.^[7] In the special case of a field F, the multiplicative group is F^\times = F \setminus \{0\}, the nonzero elements of F, since every nonzero element has an inverse and multiplication is closed on this subset.^[8] The notation (G, \times) is often used to emphasize the multiplicative operation, distinguishing it from additive groups, which employ addition with identity $0 and typically include zero but lack multiplicative inverses for it.^[9] The concept of the multiplicative group emerged within the broader development of group theory in the 19th century, following Évariste Galois's foundational work around 1830 on permutation groups under composition—a multiplicative operation—applied to the symmetries of polynomial roots.^[10] This laid the groundwork for abstract algebraic structures, where multiplicative groups became central to studying units and nonzero elements in rings and fields.^[11]

Properties

The multiplicative group of a ring R, denoted R^\times, is the set of all units in R, which are elements possessing multiplicative inverses under the ring's multiplication operation.^[12]^[13] This group structure arises directly from the ring axioms, ensuring that R^\times is closed under multiplication, contains the multiplicative identity $1, and is closed under taking inverses.^[12] For every element g \in R^\times, there exists a unique inverse g^{-1} \in R^\times such that g \cdot g^{-1} = g^{-1} \cdot g = [1](/page/1).^[12]^[14] In the specific case of a field F, the multiplicative group F^\times comprises all nonzero elements, and the inverse of g \neq 0 is explicitly given by g^{-1} = [1](/page/1)/g, leveraging the field's division property.^[14]^[15] Multiplicative subgroups of R^\times inherit the group's closure under multiplication and inverses, forming subsets that are themselves groups under the same operation.^[12] A homomorphism \phi: R^\times \to S^\times between multiplicative groups preserves the operation, satisfying \phi(g \cdot h) = \phi(g) \cdot \phi(h) for all g, h \in R^\times.^[12] Although many multiplicative groups are abelian—owing to the commutative multiplication in underlying commutative rings—non-abelian cases exist, such as the group of invertible matrices over a field.^[12] In such instances, the center Z(R^\times) = \{ z \in R^\times \mid z g = g z \ \forall g \in R^\times \} forms an abelian normal subgroup, while the commutator subgroup [R^\times, R^\times] is generated by elements of the form g h g^{-1} h^{-1}, capturing the non-commutativity.^[12]

Examples

In Fields

In fields, the multiplicative group consists of all nonzero elements, as every nonzero element has a multiplicative inverse by the field axioms. A fundamental example is the multiplicative group of nonzero rational numbers, \mathbb{Q}^\times, which under multiplication is isomorphic to \mathbb{Z}/2\mathbb{Z} \times \bigoplus_p \mathbb{Z}, where the direct sum is over all prime numbers p, the \mathbb{Z}/2\mathbb{Z} factor accounts for the sign (\{ \pm 1 \}), and each \mathbb{Z} factor corresponds to the exponent of a prime in the unique prime factorization of the absolute value of a rational number. Another familiar infinite case is the multiplicative group of nonzero real numbers, \mathbb{R}^\times. This group is isomorphic to \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}, where the isomorphism maps each nonzero real x to (\log |x|, \operatorname{sgn}(x)), with \mathbb{R} under addition corresponding to the magnitudes via the exponential map and \mathbb{Z}/2\mathbb{Z} capturing the sign.^[16] For the complex numbers, the multiplicative group \mathbb{C}^\times of nonzero elements admits a decomposition reflecting both magnitude and argument in polar form: every z \in \mathbb{C}^\times can be written as z = r e^{i\theta} with r > 0 and \theta \in [0, 2\pi), yielding an algebraic isomorphism \mathbb{C}^\times \cong \mathbb{R}_{>0} \times S^1, where \mathbb{R}_{>0} is the positive reals under multiplication and S^1 is the circle group of unit-modulus complex numbers under multiplication. Topologically, this endows \mathbb{C}^\times with the structure of a cylinder, \mathbb{R} \times S^1./04:_Cyclic_Groups/4.02:_Multiplicative_Group_of_Complex_Numbers) Finite fields provide examples of finite multiplicative groups. For a finite field \mathbb{F}_q of order q = p^k where p is prime and k \geq 1, the multiplicative group \mathbb{F}_q^\times has order q - 1 and is cyclic, meaning it is generated by a single primitive element \alpha such that every nonzero element is a power of \alpha.^[17]

In Rings

In non-field rings, the multiplicative group consists of the units, which are elements possessing multiplicative inverses within the ring, contrasting with fields where every nonzero element is invertible. A fundamental example is the ring of integers \mathbb{Z}, where the units are \{\pm 1\}, forming a cyclic group of order 2 under multiplication.^[18] Another illustrative case is the ring of Gaussian integers \mathbb{Z}, comprising elements a + bi with a, b \in \mathbb{Z} and i = \sqrt{-1}. The units here are \{\pm 1, \pm i\}, which constitute a cyclic group of order 4 generated by i.^[19] In the modular ring \mathbb{Z}/n\mathbb{Z}, the multiplicative group (\mathbb{Z}/n\mathbb{Z})^\times comprises the residue classes coprime to n. For n = p a prime, this group has order p-1. For composite n with prime factorization n = p_1^{k_1} \cdots p_r^{k_r}, the Chinese Remainder Theorem implies that (\mathbb{Z}/n\mathbb{Z})^\times is isomorphic to the direct product \prod_{i=1}^r (\mathbb{Z}/p_i^{k_i}\mathbb{Z})^\times.^[20] The order of this group is given by Euler's totient function \phi(n) = n \prod_{p \mid n} (1 - 1/p), where the product runs over distinct primes dividing n.^[21] For polynomial rings over a field, consider k where k is a field. The units are precisely the nonzero constant polynomials, isomorphic to the multiplicative group k^\times. This follows from the degree additivity under multiplication: if fg = 1, then \deg(f) + \deg(g) = 0, so both are constants.^[22]

Structure

Finite Multiplicative Groups

A fundamental result in group theory states that every finite subgroup of the multiplicative group of a field is cyclic.^[23] This theorem applies in particular to the multiplicative group of the complex numbers \mathbb{C}^\times, where all finite subgroups are cyclic groups generated by roots of unity.^[23] In the context of finite fields, the multiplicative group \mathbb{F}_q^\times of a field \mathbb{F}_q with q elements is cyclic of order q-1.^[2] For example, as seen in the case of prime fields \mathbb{F}_p^\times, this cyclicity ensures the existence of primitive elements that generate the entire group.^[24] The structure of the multiplicative group of units modulo n, denoted (\mathbb{Z}/n\mathbb{Z})^\times, is also well-understood for finite n. This group is cyclic precisely when n = 1, 2, 4, p^k, or $2p^k for an odd prime p and integer k \geq 1.^[25] In general, by the Chinese Remainder Theorem, if n = n_1 n_2 \cdots n_r where the n_i are pairwise coprime, then (\mathbb{Z}/n\mathbb{Z})^\times \cong \prod_{i=1}^r (\mathbb{Z}/n_i\mathbb{Z})^\times, yielding a direct product of cyclic groups in the non-cyclic cases.^[25] A key consequence of the cyclicity of \mathbb{F}_p^\times for prime p is Fermat's Little Theorem, which asserts that if g is not divisible by p, then g^{p-1} \equiv 1 \pmod{p}.^[24] More broadly, in any cyclic multiplicative group G of order m, every element x \in G satisfies the equation

x^m = 1,

since the order of x divides m.^[23]

Infinite Multiplicative Groups

The multiplicative group of the nonzero rational numbers, \mathbb{Q}^\times, is isomorphic to \mathbb{Z}/2\mathbb{Z} \times \bigoplus_{p} \mathbb{Z}, where the direct sum is taken over all prime numbers p.^[26] This structure arises because every nonzero rational can be uniquely written as \pm \prod p^{v_p}, with v_p \in \mathbb{Z} and only finitely many nonzero, where the \mathbb{Z}/2\mathbb{Z} factor corresponds to the sign and each \mathbb{Z} factor to the p-adic valuation. Consequently, \mathbb{Q}^\times is a free abelian group of countable rank, reflecting the countably infinite set of primes.^[27] The multiplicative group of the nonzero real numbers, \mathbb{R}^\times, is isomorphic to \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}.^[28] The \mathbb{Z}/2\mathbb{Z} factor accounts for the sign, while the positive reals \mathbb{R}_{>0}^\times under multiplication are isomorphic to the additive group (\mathbb{R}, +) via the exponential map, or equivalently the logarithm map provides the isomorphism \log: \mathbb{R}_{>0}^\times \to (\mathbb{R}, +). This isomorphism highlights the vector space structure over \mathbb{Q} of both groups, though \mathbb{R}^\times has uncountable rank. The multiplicative group of the nonzero complex numbers, \mathbb{C}^\times, is a Lie group of real dimension 2. Every element can be expressed in polar form as r e^{i\theta} with r > 0 and \theta \in [0, 2\pi), and the group operation corresponds to componentwise addition in the logarithm: multiplication becomes (r_1 e^{i\theta_1}) (r_2 e^{i\theta_2}) = (r_1 r_2) e^{i(\theta_1 + \theta_2)}. This endows \mathbb{C}^\times with the structure of a direct product of the positive reals under multiplication and the circle group S^1 = \{ e^{i\theta} \mid \theta \in \mathbb{R} \}.^[29] In infinite multiplicative groups such as these, the torsion subgroup—consisting of elements of finite order—plays a key role. For \mathbb{Q}^\times and \mathbb{R}^\times, the torsion subgroup is \{ \pm 1 \} \cong \mathbb{Z}/2\mathbb{Z}. In \mathbb{C}^\times, the torsion elements are precisely the roots of unity, forming a countable torsion subgroup isomorphic to the direct limit of cyclic groups \mathbb{Z}/n\mathbb{Z} over all positive integers n.^[30] Unlike finite multiplicative groups, which admit a complete classification via cyclic decompositions, there is no general classification theorem for infinite multiplicative groups. These groups can exhibit highly diverse structures, often requiring advanced invariants like rank or divisibility properties for partial understanding. However, certain subgroups, such as the positive reals \mathbb{R}_{>0}^\times, are divisible: for any element x and integer n \geq 1, there exists y such that y^n = x, reflecting their isomorphism to vector spaces over \mathbb{Q}.^[31]

Special Cases

Roots of Unity

The nth roots of unity are the complex numbers \zeta satisfying \zeta^n = 1, explicitly given by \zeta_k = e^{2\pi i k / n} for integers k = 0, 1, \dots, n-1. Under complex multiplication, these form a finite abelian group \mu_n of order n, which is cyclic and generated by the primitive nth root of unity e^{2\pi i / n}.^[32]^[33] Among the elements of \mu_n, the primitive nth roots of unity are those of exact order n, numbering \phi(n), where \phi is Euler's totient function. The nth cyclotomic polynomial

\Phi_n(x) = \prod_{\substack{k=1 \\ \gcd(k,n)=1}}^n \left( x - e^{2\pi i k / n} \right)

is the monic polynomial of degree \phi(n) whose roots are precisely the primitive nth roots of unity; it is irreducible over \mathbb{Q} and serves as the minimal polynomial of any primitive nth root over \mathbb{Q}.^[34]^[33] The group of all roots of unity is the union \mu_\infty = \bigcup_{n=1}^\infty \mu_n over the natural numbers n, taken as a direct limit under the natural inclusions \mu_m \hookrightarrow \mu_n whenever m divides n. This group \mu_\infty coincides with the torsion subgroup of the multiplicative group \mathbb{C}^\times of nonzero complex numbers, consisting of all elements of finite order, and is isomorphic (as an abelian group) to the additive group \mathbb{Q}/\mathbb{Z}.^[35]^[33] In Galois theory, adjoining a primitive nth root of unity \zeta_n to \mathbb{Q} yields the cyclotomic field \mathbb{Q}(\zeta_n), a Galois extension of \mathbb{Q} whose Galois group is isomorphic to the multiplicative group of units modulo n, that is, \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times. This isomorphism arises from the action of Galois automorphisms on \zeta_n via \sigma_a(\zeta_n) = \zeta_n^a for a \in (\mathbb{Z}/n\mathbb{Z})^\times.^[36]^[33]

Units in Integral Domains

In integral domains, particularly the rings of integers in number fields, the multiplicative group of units plays a central role in algebraic number theory, capturing the invertible elements and influencing ideal structures and factorization properties.^[37] For a number field K with ring of integers \mathcal{O}_K, the unit group \mathcal{O}_K^\times consists of elements \alpha \in \mathcal{O}_K such that there exists \beta \in \mathcal{O}_K with \alpha \beta = 1. This group is finitely generated, as established by Dirichlet's unit theorem.^[38] Dirichlet's unit theorem provides the precise structure: if K has r_1 real embeddings and r_2 pairs of complex conjugate embeddings, then \mathcal{O}_K^\times \cong \mathbb{Z}^{r_1 + r_2 - 1} \times \mu, where \mu is the finite torsion subgroup consisting of the roots of unity in \mathcal{O}_K.^[38] The rank of the free abelian part is r = r_1 + r_2 - 1, reflecting the number of independent units of infinite order.^[37] This decomposition arises from the logarithmic embedding of units into \mathbb{R}^{r_1 + r_2}, where the image lies in a hyperplane, ensuring finite generation.^[38] In real quadratic fields, such as K = \mathbb{Q}(\sqrt{d}) for square-free positive integer d, the unit group has rank 1, leading to infinite order elements connected to Pell's equation x^2 - d y^2 = \pm 1. Solutions to this equation yield units in \mathbb{Z}[\sqrt{d}], with the fundamental unit \varepsilon generating the infinite cyclic part via powers \varepsilon^n for n \in \mathbb{Z}.^[38] For example, in \mathbb{Z}[\sqrt{2}], the fundamental unit is $1 + \sqrt{2}, and the full unit group is \{\pm (1 + \sqrt{2})^n \mid n \in \mathbb{Z}\}.^[39] The unit group and the class number— the order of the ideal class group measuring deviation from unique factorization—together characterize the arithmetic of \mathcal{O}_K in algebraic number theory. While the unit group describes invertible elements and principal ideals, the class number quantifies non-principal ideals, with both influencing computations in class field theory and regulator estimates.^[37]

References

[1]
group of units in nLab
### Summary of Group of Units (nLab)
[2]
[PDF] The Group of Units in the Integers mod n
Feb 22, 2018 · The group of units (Un) in Zn are elements with multiplicative inverses, forming a group under multiplication mod n. For example, U11 = {1, 2, ...
[3]
[PDF] The Multiplicative Group of a Finite Field
The multiplicative group of a finite field, F× = F \ {0}, is a cyclic group under multiplication.
[4]
AATA Multiplicative Group of Complex Numbers
Every nonzero complex number z = a + b i has a multiplicative inverse; that is, there exists a z − 1 ∈ C ∗ such that . z z − 1 = z − 1 z = 1 .
[5]
Multiplicative Group -- from Wolfram MathWorld
A multiplicative group is a group where the group operation is multiplication, denoted by a dot or omitted, and the identity is 1.Missing: abstract | Show results with:abstract
[6]
[PDF] Standard definitions for rings - Keith Conrad
This says a ring is a commutative group under addition, it is a “group without inverses” under multiplication, and multiplication distributes over addition.
[7]
[PDF] 1. Rings: definitions, examples, and basic properties - UCSD Math
seen to be a group under the multiplication operation of the ring. It is called the units group of R. Another common notation for this group is R×.
[8]
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_(Clark)/01%3A_Chapters/1.05%3A_The_Group_of_Units
[9]
[PDF] The Very Basics of Groups, Rings, and Fields
A RING is a GROUP under addition ... Examples: Z/pZ is a field, since Z/pZ is an additive group and (Z/pZ) − {0} = (Z/pZ)× is a group under multiplication.
[10]
[PDF] The Evolution of Group Theory: A Brief Survey - Israel Kleiner
Mar 14, 2004 · Galois was the first to use the term "group" in a technical sense- to him it signified a collection of permutations closed under multiplication: ...
[11]
[PDF] math 101a: algebra i part b: rings and modules - Brandeis
Oct 10, 2007 · group rings. Suppose that G is a multiplicative group and R is a commutative ring. Then the group ring RG is defined to be the set of all ...
[12]
[PDF] Lecture Notes on Abstract Algebra - Stephen Doty
Nov 21, 2024 · order in the multiplicative group of units in a ring R. Then ⟨a⟩ = {ak | k ∈. Z} is an infinite cyclic group isomorphic to the additive ...
[13]
[PDF] Chapter 8: Rings - Mathematical and Statistical Sciences
A unit is any u ∈ R that has a multiplicative inverse: some v ∈ R such that uv = vu = 1. Let U(R) be the set (a multiplicative group) of units of R. Proposition.Missing: abstract | Show results with:abstract
[14]
[PDF] Math 4310 Handout - Fields - Cornell Mathematics
9. (Multiplicative inverses): For each nonzero a ∈ F there exists an element a−1 ∈ F such that a·a−1 = 1. We then define subtraction of two elements by a − b = ...
[15]
[PDF] Notes on Fields
One can show, using properties of prime numbers, that any nonzero element of Zp has a multiplicative inverse, so that Zp is in fact a field. It may be helpful ...
[16]
[PDF] The First Isomorphism Theorem
Mar 22, 2018 · R∗. {1, −1}. ≈ R+. R∗ is the group of nonzero real numbers under multiplication. R+ is the group of positive real numbers under multiplication.
[17]
[PDF] 3 Finite fields and integer arithmetic - MIT Mathematics
Feb 13, 2019 · The multiplicative group of a finite field is cyclic. If α is a generator for the multiplicative group F× q , then it generates Fq as an ...
[18]
[PDF] THE ARITHMETIC OF NUMBER RINGS Peter Stevenhagen
The ring Z of 'ordinary' integers lies at the very root of number theory, and when studying its properties, the concept of divisibility of integers naturally ...
[19]
[PDF] THE GAUSSIAN INTEGERS Since the work of Gauss, number ...
Knowing a Gaussian integer up to multiplication by a unit is analogous to knowing an integer up to its sign. While there is no such thing as inequalities on ...
[20]
[PDF] The Chinese Remainder Theorem
Feb 19, 2018 · Remark 1. The Chinese remainder theorem (CRT) asserts that there is a unique class a + NZ so that x solves the system (2) if and only if x ∈ a ...
[21]
[PDF] 4 Euler's Totient Function
If k ≥ 1 is such that ak ≡ 1 (mod n), then gcd(a, n) = 1 (a is a unit modulo n). The proof is a (hopefully) straightforward exercise. We turn now to the ...
[22]
[PDF] Nilpotents, units, and zero divisors for polynomials - Keith Conrad
The units are the units in A since deg(fg) = deg f + deg g when f,g 6= 0, so fg = 1 ⇒ deg f,deg g = 0 ⇒ f ∈ A×, and the converse is obvious. The only zero ...
[23]
[PDF] Finite Multiplicative Subgroups of a Field
Let G ⊂ F∗ be a finite group. There are several ways to prove that G is cyclic. All proofs are based on the fact that the equation xd = 1 can have at most ...<|separator|>
[24]
[PDF] CYCLICITY OF (Z/(p)) 1. Introduction For a prime p, the group (Z/(p ...
Theorem 4.1. For each prime p, the group (Z/(p))× is cyclic. Proof. Let n be the maximal order among the elements in (Z/(p))×.
[25]
[PDF] MULTIPLICATIVE GROUPS IN Zm 1. Abstract Our goal will be to find ...
It is the set of numbers less than n and relatively prime to n under the operation multiplication modulo n. Page 4.
[26]
How to find the index of this subgroup of Q - Math Stack Exchange
Oct 1, 2016 · ... primes 3 modulo 4, gives the decomposition. Q×≅Z/2Z⊕⨁pZ. where the direct sum runs over all primes p. Since there are infinitely many primes ...Group Q∗ as direct product/sum - Math Stack ExchangeProve that Q× not isomorphic to Zn - Math Stack ExchangeMore results from math.stackexchange.com
[27]
Is $<\mathbb Q^+, \times>$ the free abelian group on countably ...
Aug 7, 2014 · The free abelian group generated by a set does not include infinite products of elements of the set, only finite products, so there is no ...Free abelian group - exercise - Mathematics Stack ExchangeIs (Q>0,*) a free abelian group with countable basis?More results from math.stackexchange.com
[28]
Multiplicative groups of nonzero rational and real numbers
Feb 6, 2018 · Multiplicative groups of nonzero rational and real numbers ... Two structures cannot be isomorphic unless they both have the same cardinality.Nonzero rationals under multiplication are not a cyclic groupProve that none of the isomorphisms above can be extended to an ...More results from math.stackexchange.com
[29]
IAAWA Multiplicative Group of Complex Numbers - UTK Math
Every nonzero complex number z = a + b i has a multiplicative inverse; that is, there exists a z − 1 ∈ C such that . z z − 1 = z − 1 z = 1 .
[30]
Is a field uniquely determined by its multiplicative group/how much ...
Apr 23, 2010 · Q∗ is isomorphic to {±1} times a free abelian group of countable rank. The same is true for an imaginary quadratic field of class number 1 and ...Non-split extension of the rationals by the integers - MathOverflowDo the algebraic integers form a free abelian group? - MathOverflowMore results from mathoverflow.net
[31]
[PDF] Divisible multiplicative groups of fields - UCCS Faculty Sites
We begin this section with a determination of the divisible abelian groups G which can be realized as the multiplicative group of an absolutely algebraic field ...<|control11|><|separator|>
[32]
Root of Unity -- from Wolfram MathWorld
### Summary of nth Roots of Unity
[33]
[PDF] 19. Roots of unity
An element ω ∈ k× is a primitive nth root of unity in k if and only if ω is an element of order n in the group µn of all nth roots of unity in k. If so ...
[34]
Cyclotomic Polynomial -- from Wolfram MathWorld
The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials. The first few ...
[35]
https://ysharifi.wordpress.com/2019/10/05/the-additive-group-q-z/
[36]
[PDF] cyclotomic extensions - keith conrad
The important algebraic fact we will explore is that cyclotomic extensions of every field have an abelian Galois group; we will look especially at cyclotomic ...
[37]
[PDF] Algebraic Number Theory - James Milne
the ring of integers in the number field, the ideals and units in the ring of.
[38]
[PDF] dirichlet's unit theorem - keith conrad
Q with a unique real root α and Z[α]× = ±αZ (even if Z[α] is not the integers of Q(α)). By the rational roots theorem, a rational root of f(T) must be ±1 ...
[39]
Fundamental Unit -- from Wolfram MathWorld
The fundamental units for real quadratic fields Q(sqrt(D)) may be computed ... 2+sqrt(3), 62, 63+8sqrt(62). 13, 1/2(3+sqrt(13)), 63, 8+3sqrt(7). 14, 15+4sqrt ...