Hensel’s proof of the transcendence of e
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Hensel proof

In 1905, Hensel provided the following proof about the transcendence of \(e\).
By the Taylor expansion of \(e\) at \(0\) we get:

\[e^p = \sum_{n=0}^{\infty} \frac{p^n}{n!} = 1 + p \sum_{n=1}^{\infty} \frac{p^{n-1}}{n!},\]

From this expansion, we see that \(e\) satisfies an equation of the form \(y^p = 1 + pu\), where \(u\) is a \(p\)-adic unit (Exercise).

But the polynomial \(y^p - (1 + pu)\) is irreducible over \(\mathbb{Q}_p\), the fraction field of \(\mathbb{Z}_p\):
writing \(y = 1 + z\) and expanding, we get

\[z^p + pz^{p-1} + \dots + pz - pu\]

which is irreducible by the Eisenstein criterion.
Therefore \(e\) must have degree at least \(p\) over \(\mathbb{Q}_p\). Hence:

\[[\mathbb{Q}(e) : \mathbb{Q}] \ge p,\]

but this is true for any prime \(p\), so \(e\) must be transcendental.

Now, can you spot the flaw in this argument?