PlanetMath: prime residue class

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prime residue class

(Definition)

Let be a positive integer. There are residue classes $a\!+\!m\mathbb{Z}$ modulo . Such of them which have

$\displaystyle \gcd(a,\,m) = 1,$

are called the prime residue classes or prime classes modulo , and they form an Abelian group with respect to the multiplication

$\displaystyle (a\!+\!m\mathbb{Z})\!\cdot\!(b\!+\!m\mathbb{Z}) := ab\!+\!m\mathbb{Z}.$

This group is called the residue class group modulo . Its order is $\varphi(m)$ , where $\varphi$ means Euler's totient function. For example, the prime classes modulo 8 (i.e. $1\!+\!8\mathbb{Z}$ , $3\!+\!8\mathbb{Z}$ , $5\!+\!8\mathbb{Z}$ , $7\!+\!8\mathbb{Z}$ ) form a group isomorphic to the Klein 4-group.

The prime classes are the units of the residue class ring $\mathbb{Z}/m\mathbb{Z} = \mathbb{Z}_m$ consisting of all residue classes modulo .

Analogically, in the ring of integers of any algebraic number field, there are the residue classes and the prime residue classes modulo an ideal $\mathfrak{a}$ of . The number of all residue classes is N $(\mathfrak{a})$ and the number of the prime classes is also denoted by $\varphi(\mathfrak{a})$ . It may be proved that

$\displaystyle \varphi(\mathfrak{a}) =$ N $\displaystyle (\mathfrak{a})\prod_{\mathfrak{p}\vert\mathfrak{a}}\left(1-\frac{1}{\mbox{N}(\mathfrak{p})}\right);$

N is the absolute norm of ideal and $\mathfrak{p}$ runs all distinct prime ideals dividing $\mathfrak{a}$ (cf. the first formula in the entry “Euler phi function”). Moreover, one has the result

$\displaystyle \alpha^{\varphi(\mathfrak{a})} \equiv 1 \pmod{\mathfrak{a}}$

for $((a),\,\mathfrak{a}) = (1)$ , generalising the Euler-Fermat theorem.

"prime residue class" is owned by pahio.

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Other names:

prime class

Also defines:

residue class group

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Cross-references: Euler-Fermat theorem, prime ideals, absolute norm of ideal, number, ideal, algebraic number field, residue class ring, units, Klein 4-group, isomorphic, Euler's totient function, order, group, multiplication, abelian group, residue classes, integer, positive
There are 2 references to this entry.

This is version 13 of prime residue class, born on 2006-03-03, modified 2008-05-28.
Object id is 7668, canonical name is PrimeResidueClass.
Accessed 2911 times total.

Classification:

AMS MSC:	11A07 (Number theory :: Elementary number theory :: Congruences; primitive roots; residue systems)
	13M99 (Commutative rings and algebras :: Finite commutative rings :: Miscellaneous)
	20K01 (Group theory and generalizations :: Abelian groups :: Finite abelian groups)

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