Congruency grows complex

Number Theory Level 5

If you consider the Gaussian integers \(\mathbb{Z}[i]\) modulo \(2+i\), how many congruency classes are there?


  • Gaussian integers are of the form \(a+bi\) where \(a\) and \(b\) are integers.

  • \(z_1\equiv z_2 \pmod{2+i}\) means that \(z_1-z_2=(2+i)w\) for some Gaussian integer \(w\).

Bonus Question: How many of the congruency classes are units, meaning that they have a multiplicative inverse?


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