Wilson's theorem (Q988061)

More than two hundred years ago Gauss generalized Wilson's theorem (\((p-1)!\equiv -1 \pmod p\)), that is he gave the following proposition: Proposition 1. (Gauss)\newline The product of all elements in \((\mathbb Z/A\mathbb Z)^\times\) is \(\bar 1\) or \(-\bar 1\). Later \textit{K. Hensel} [J. Reine Angewandte Math. 146, 189--215 (1916; JFM 46.0251.01)] developed his local notions, which could have allowed him to extend the result from \(\mathbb Z\) to rings of integers in number fields. Proposition 2. For an ideal \(\mathfrak{a}\subset \mathfrak{o}\) in the ring of integers of a number field \(K\), the product of all elements in \((\mathfrak{o}/\mathfrak{a})^\times\) is \(\bar 1\), except that it is\newline (1) \(-\bar 1\) when \(\mathfrak{a}\) has precisely one odd prime divisor, and \(v_{\mathfrak{p}}(\mathfrak{a})<2\) for every even prime ideal \(\mathfrak p\),\newline (2) \(\bar 1 + \bar\pi\) (resp \(\bar 1+ \bar\pi^2\)) when all prime divisors of \(\mathfrak a\) are even and for precisely one of them, say \(\mathfrak p\), \(v_{\mathfrak p}(\mathfrak a)>1\) with moreover \(v_{\mathfrak p}(\mathfrak a)=2\), \(f_{\mathfrak p}=1\) (resp. \(v_{\mathfrak p}(\mathfrak a)=3\), \(f_{\mathfrak p}=1\), \(e_{\mathfrak p}>1\)); here \(\pi\) is any element of \(\mathfrak p\) not in \(\mathfrak p^2\), and we have identified \((\mathfrak o/\mathfrak p^2)^\times\) (resp. \((\mathfrak o/\mathfrak p^3)^\times\)) with a subgroup of \((\mathfrak o/\mathfrak a)^\times\). In the proposition above \(v_{\mathfrak p}(\mathfrak a)\) is the exponent of \(\mathfrak p\) in the prime decomposition of \(\mathfrak a\); \(f_{\mathfrak p}\) is the residual degree and \(e_{\mathfrak p}\) the ramification index of \(K_{\mathfrak p}|\mathbb Q_p\) (\(p\) being the rational prime which belongs to \(\mathfrak p\)). The author's goal is to show how Hensel could have done it. For proving this the author uses the following proposition: Proposition 3. Denoting by \(e\) the ramification index and by \(f\) the residual degree of \(K|\mathbb Q_p\), we have \(d_2((\mathfrak o/\mathfrak p^n)^\times)=\)\newline (1) 1 if \(p\neq 2\), \newline (2) 0 if \(p=2\), \(n=1\),\newline (3) 1 if \(p=2\), \(n=2\), \(f=1\),\newline (4) 1 if \(p=2\), \(n=3\), \(f=1\), \(e>1\),\newline (5) \(>1\) in all other cases.\newline For any \(\mathfrak o\)-basis \(\pi\) of \(\mathfrak p\), the unique order-2 element in the cases \(d_2=1\) is \newline (1) \(-\bar 1\) if \(p\neq 2\),\newline (2) \(\bar 1+\bar \pi\) if \(p=2\), \(n=2\), \(f=1\),\newline (3) \(\bar 1+\bar \pi^2\) if \(p=2\), \(n=3\), \(f=1\), \(e>1\). This proof of Proposition 2 is shorter, simpler, more direct and more conceptual than \textit{M. Laššák}'s one [Math. Slovaca 50, No. 3, 303--314 (2000; Zbl 0997.11086)].