Nontrivial Quantization of $\phi^4_n$, $n\ge2$

Conventional quantization of covariant scalar field models $\phi^4_n$, for spacetime dimensions $n\ge5$ are trivial, and this may also be true for $n=4$ as well. However, an alternative ${\cal O}(\hbar)$ counterterm leads to nontrivial results for all $n\ge4$, and provides a different quantization for $n=2,3$ as well. In this article we determine the counterterm that provides these desirable properties as simply and directly as possible. The same counterterm also resolves models such as $\phi^p_n$ for all even $p$, including those where $p>2n/(n-2)$, which are traditionally regarded as nonrenormalizable.