Exercise 15.5.10

Proof. By Theorem 15.2.5, since $n-1$ is even, for all $x\in ℝ$,

Moreover, by (15.9),

Applying these two formula with $x=\frac{\varpi }{n}$, we obtain

If $n=1$, then $\phi \left(\frac{\varpi }{n}\right)=0$, but in this case ${\phi }^{\prime }(\varpi )=-1\in \mathbb{Q}\subset L$.

If $n>1$, then $\phi \left(\frac{\varpi }{n}\right)\ne 0$, thus the last equality shows that $P_{n-1}\left({\phi }^4\left(\frac{\varpi }{n}\right)\right)\ne 0$. We obtain

This equality shows that

For all $n\in \mathbb{Z},n>0$,

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