galois.is_cyclic

class galois.is_cyclic(n)

Determines whether the multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic.

The multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is the set of positive integers \(1 \le a < n\) that are coprime with \(n\). \((\mathbb{Z}/n\mathbb{Z}){^\times}\) being cyclic means that some primitive root of \(n\), or generator, \(g\) can generate the group \(\{g^0, g^1, g^2, \dots, g^{\phi(n)-1}\}\), where \(\phi(n)\) is Euler’s totient function and calculates the order of the group. If \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic, the number of primitive roots is found by \(\phi(\phi(n))\).

\((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic if and only if \(n\) is \(2\), \(4\), \(p^k\), or \(2p^k\), where \(p\) is an odd prime and \(k\) is a positive integer.

Parameters

n (int) – A positive integer.

Returns

True if the multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic.

Return type

bool

Examples

The elements of \((\mathbb{Z}/n\mathbb{Z}){^\times}\) are the positive integers less than \(n\) that are coprime with \(n\). For example, \((\mathbb{Z}/14\mathbb{Z}){^\times} = \{1, 3, 5, 9, 11, 13\}\).

# n is of type 2*p^k, which is cyclic
In [486]: n = 14

In [487]: galois.is_cyclic(n)
Out[487]: True

# The congruence class coprime with n
In [488]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[488]: {1, 3, 5, 9, 11, 13}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [489]: phi = galois.euler_totient(n); phi
Out[489]: 6

In [490]: len(Znx) == phi
Out[490]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [491]: for a in Znx:
   .....:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   .....:     primitive_root = span == Znx
   .....:     print("Element: {:2d}, Span: {:<20}, Primitive root: {}".format(a, str(span), primitive_root))
   .....: 
Element:  1, Span: {1}                 , Primitive root: False
Element:  3, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  5, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  9, Span: {9, 11, 1}          , Primitive root: False
Element: 11, Span: {9, 11, 1}          , Primitive root: False
Element: 13, Span: {1, 13}             , Primitive root: False

In [492]: roots = galois.primitive_roots(n); roots
Out[492]: [3, 5]

# Euler's totient function phi(phi(n)) counts the primitive roots of n
In [493]: len(roots) == galois.euler_totient(phi)
Out[493]: True

A counterexample is \(n = 15 = 3*5\), which doesn’t fit the condition for cyclicness. \((\mathbb{Z}/15\mathbb{Z}){^\times} = \{1, 2, 4, 7, 8, 11, 13, 14\}\).

# n is of type p1^k1 * p2^k2, which is not cyclic
In [494]: n = 15

In [495]: galois.is_cyclic(n)
Out[495]: False

# The congruence class coprime with n
In [496]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[496]: {1, 2, 4, 7, 8, 11, 13, 14}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [497]: phi = galois.euler_totient(n); phi
Out[497]: 8

In [498]: len(Znx) == phi
Out[498]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [499]: for a in Znx:
   .....:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   .....:     primitive_root = span == Znx
   .....:     print("Element: {:2d}, Span: {:<13}, Primitive root: {}".format(a, str(span), primitive_root))
   .....: 
Element:  1, Span: {1}          , Primitive root: False
Element:  2, Span: {8, 1, 2, 4} , Primitive root: False
Element:  4, Span: {1, 4}       , Primitive root: False
Element:  7, Span: {1, 4, 13, 7}, Primitive root: False
Element:  8, Span: {8, 1, 2, 4} , Primitive root: False
Element: 11, Span: {1, 11}      , Primitive root: False
Element: 13, Span: {1, 4, 13, 7}, Primitive root: False
Element: 14, Span: {1, 14}      , Primitive root: False

In [500]: roots = galois.primitive_roots(n); roots
Out[500]: []

# Note the max order of any element is 4, not 8, which is Carmichael's lambda function
In [501]: galois.carmichael(n)
Out[501]: 4