On the Arithmetic of Endomorphism Ring End $$({Z_p} \times {Z_{{p^m}}})$$ ( Z p amp;#x00D7; Z p m )
Xiusheng Liu, Hualu Liu, Peng Hu · Wuhan University Journal of Natural Sciences · 2018
For a prime p, let $${E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}} aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}$$ . We first establish a ring isomorphism from $${Z_{p,{p^m}}}$$ onto $${E_{p,{p^m}}}$$ . Then we provide a way to compute -d and d–1 by using arithmetic in Zp and $${Z_{{p^m}}}$$ , and characterize the invertible elements of $${E_{p,{p^m}}}$$ . Moreover, we introduce the minimal polynomial for each element in $${E_{p,{p^m}}}$$ and give its applications.