Thursday, 27 January 2022

On generators of multiplicative group

 Generators of multiplicative group \((\mathbb{Z}/2^{\alpha}\mathbb{Z})^*\)

The set of all unit elements in the cyclic group \(\mathbb{Z}/N\mathbb{Z}\) is a group under multiplication modulo \(N\) of order \(\phi(N)\), where \(\phi\) is the Euler-totient function. It is denoted by \((\mathbb{Z}/N\mathbb{Z})^*\).
It is well known that \((\mathbb{Z}/N\mathbb{Z})^*\) is not always cyclic (Primitive element theorem) (See Chater 10, [1]). For example \((\mathbb{Z}/8\mathbb{Z})^*=\{1,3,5,7\}\) is not cyclic. \((\mathbb{Z}/2\mathbb{Z})^*=\{1\}\) and \((\mathbb{Z}/4\mathbb{Z})^*=\{1,3\}\) are trivially cyclic.

Question: Show that for \(\alpha\geq 3\), \((\mathbb{Z}/2^{\alpha}\mathbb{Z})^*\) is generated by \(-1\) and 5?
You may use the following sage code for validating the result:
Reference: [1] T M Apostol, Introduction to Analytics Number Theory

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