The Unit Group

UnitGroup(R): RngIntRes -> GrpAb, Map

Given \(R={{\mathbb{Z}}/m{\mathbb{Z}}}\), construct the unit group of \(R\) as an abelian group. This returns an (additive) abelian group \(A\) of order \(\phi(m)\), together with a map from \(A\) to \(R\).

IsPrimitive(n): RngIntResElt -> BoolElt

Returns true if the element \(n\in{{\mathbb{Z}}/m{\mathbb{Z}}}\) is primitive, that is, if it generates the multiplicative group of \({{\mathbb{Z}}/m{\mathbb{Z}}}\), false otherwise.

PrimitiveElement(R): RngIntRes -> RngIntResElt

PrimitiveRoot(R): RngIntRes -> RngIntResElt

Given \(R = {\mathbb{Z}}/m{\mathbb{Z}}\), this function returns a generator for the group of units of \(R\) if this group is cyclic, and returns \(0\) otherwise. Thus a valid generator is only returned if \(m = 2, 4, p^t\) or \(2p^t\), with \(p\) an odd prime and \(t\geq 1\).

Order(a): RngIntResElt -> RngIntElt

Given an element \(a\) belonging to \({{\mathbb{Z}}/m{\mathbb{Z}}}\), return the multiplicative order \(k \geq 1\) of \(a\) if \(a\) is in the unit group \({({{\mathbb{Z}}/m{\mathbb{Z}}})^*}\), and zero if \(a\) is not a unit.

Normalize(x): RngIntRes -> RngIntResElt, RngIntResElt

Normalise(x): RngIntRes -> RngIntResElt, RngIntResElt

Given an element \(x\in R = {\mathbb{Z}}/m{\mathbb{Z}}\), this function returns the unique canonical associate \(y\in R\) of \(x\) and a unit \(u\in R\) such that \(u\cdot x = y\). The canonical associate of \(x\) is the GCD of \(x\) and \(m\), considered as natural integers (unless \(x\) is 0, in which case it is 0).

Example: Unit Group (ex-8ae5da)

We determine the unit group of the ring with modulus \(735\) and then verify its order by comparing it with \(\phi(m)\).

> m := 735;
> R := ResidueClassRing(m);
Residue class ring of integers modulo 735
> U, psi := UnitGroup(R);
> U;
Abelian Group isomorphic to Z/2 + Z/2 + Z/84
Defined on 3 generators
Relations:
    2*U.1 = 0
    4*U.2 = 0
    42*U.3 = 0
> #U;
336
> EulerPhi(735);
336

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So the order of \(U\) is equal to \(\phi(m)\) as it should be. Finally, we look for three elements of \(R\) that generate the unit group.

> gens := [ psi(U.i) : i in [1..3] ]; gens;
> [ Order(x) : x in gens ];
[ 2, 4, 42 ]

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Example: Cyclic Unit Group (ex-bde8d2)

We construct a residue class ring \(R = {\mathbb{Z}}/m{\mathbb{Z}}\) having cyclic unit group. By a theorem of Gauss, the ring \(R\) has cyclic unit group precisely when \(n = 4, n = p^e,\) or \(n = 2p^e\), and \(p\) is an odd prime.

> R := IntegerRing(50);
> U, psi := UnitGroup(R);
Abelian Group isomorphic to Z/20
Defined on 1 generator
Relations:
    20*U.1 = 0
> w := PrimitiveElement(R);
> w;
3
> Order(w);
20

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We verify that the powers of \(w\) are precisely the elements of the unit group \(U\).

> powers := { w^i : i in [0..19] };
> powers;
{ 29, 1, 31, 3, 33, 7, 37, 9, 39, 11, 41, 13, 43, 17, 47, 19, 49, 21, 23, 27 }
> powers eq { psi(u) : u in U };
true

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