TITLE: How are polynomials mod m reduced?
QUESTION [6 upvotes]: How do you reduce polynomials that are mod m?
For example if I have 10x + 5 (mod 3) can I just reduce that to x + 2 (mod 3)?

REPLY [7 votes]: If $\rm\ m\in \mathbb Z\ $ then $\rm\ f\:\equiv\: g\ \ (mod\ m\: \mathbb Z[x])\:$ means $\rm\:f-g\:\in\: m\:\mathbb Z[x]\:,\:$ i.e. $\rm\:m\ |\ f-g\ $ in $\rm\:\mathbb Z[x]\:.\:$ This is equivalent to saying that $\rm\:m\ |\ f_{\:i} - g_{\:i}\:,\:$ i.e. the polynomials have equivalent coefficients $\rm\:(mod\ m)\:.$  
Said structurally $\rm\ \mathbb Z[x]/m\:\mathbb Z[x]\ \cong \mathbb (\mathbb Z/m\mathbb Z)[x]\:.$