I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping.
$$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ $$H^\prime=k+kt^{4\nu}(1+t+t^2)+kt^{6\nu}(1+t+t^2)+kt^{7\nu}(1+t+t^2)+k[[t]]t^{8\nu}$$