  We revisit the effective field theory of the standard model that is extended
with sterile neutrinos, $N$. We examine the basis of complete and independent
effective operators involving $N$ up to mass dimension seven (dim-7). By
employing equations of motion, integration by parts, and Fierz and group
identities, we construct relations among operators that were considered
independent in the previous literature, and find seven redundant operators at
dim-6, sixteen redundant operators and two new operators at dim-7. The correct
numbers of operators involving $N$ are, without counting Hermitian conjugates,
$16~(L\cap B)+1~(\slashed{L}\cap B)+2~(\slashed{L}\cap\slashed{B})$ at dim-6,
and $47~(\slashed{L}\cap B)+5~(\slashed{L}\cap\slashed{B})$ at dim-7. Here
$L/B~(\slashed L/\slashed B)$ stands for lepton/baryon number conservation
(violation). We verify our counting by the Hilbert series approach for $n_f$
generations of the standard model fermions and sterile neutrinos. When
operators involving different flavors of fermions are counted separately and
their Hermitian conjugates are included, we find there are $29~(1614)$ and
$80~(4206)$ operators involving sterile neutrinos at dim-6 and dim-7
respectively for $n_f=1~(3)$.
