What does the notation $x\equiv y$ mod $2\mathbb{Z}[\zeta_3]$ mean?
What does the notation $x\equiv y$ mod $2\mathbb{Z}[\zeta_3]$ mean?
2 Answers
It means $\rm\ x - y\ =\ 2\ z\ $ for some $\rm\ z \in \mathbb Z[\zeta_3]\:,\ $ i.e. $\rm\ z = m + n\ \zeta_3,\ $ where $\rm\ \zeta_3^2 + \zeta_3 + 1 = 0\:.$
This is a generalization of the notion of congruence in the ring $\rm\ \mathbb Z/n\ $ of integers modulo $\rm\:n\:$. Recall that when solving integer problems it often proves convenient to attempt to decompose the problem into simpler problems in the finite rings/fields $\rm\ \mathbb Z/n\:.\ $ The same technique proves useful for general rings, where one consider "simpler" images $\rm\ R/I \ $ where $\rm\:I\:$ is the congruence class of $0\:,\ $ the set of all elements that are mapped to $0$ in the quotient ring. This has a fundamental structure known as an ideal, of which the above principal ideal $\rm\ n\ R\ =\ \{ n\:r\ :\ r\in R\}\ $ is a prototypical example.
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0Yes, that typo is now fixed. – 2011-02-08