## Wilson’s theorem.(English)Zbl 1204.11166

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)
The product of all elements in $$(\mathbb Z/A\mathbb Z)^\times$$ is $$\bar 1$$ or $$-\bar 1$$.
Later 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
(1) $$-\bar 1$$ when $$\mathfrak{a}$$ has precisely one odd prime divisor, and $$v_{\mathfrak{p}}(\mathfrak{a})<2$$ for every even prime ideal $$\mathfrak p$$,
(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 $$d_2((\mathfrak o/\mathfrak p^n)^\times)=$$
(1) 1 if $$p\neq 2$$,
(2) 0 if $$p=2$$, $$n=1$$,
(3) 1 if $$p=2$$, $$n=2$$, $$f=1$$,
(4) 1 if $$p=2$$, $$n=3$$, $$f=1$$, $$e>1$$,
(5) $$>1$$ in all other cases.
For any $$\mathfrak o$$-basis $$\pi$$ of $$\mathfrak p$$, the unique order-2 element in the cases $$d_2=1$$ is
(1) $$-\bar 1$$ if $$p\neq 2$$,
(2) $$\bar 1+\bar \pi$$ if $$p=2$$, $$n=2$$, $$f=1$$,
(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 M. Laššák’s one [Math. Slovaca 50, No. 3, 303–314 (2000; Zbl 0997.11086)].

### MSC:

 11R04 Algebraic numbers; rings of algebraic integers 11R44 Distribution of prime ideals

### Keywords:

Wilson’s theorem; prime number; prime ideal; finite extension

### Citations:

JFM 46.0251.01; Zbl 0997.11086
Full Text:

### References:

 [1] C. Gauss, Disquisitiones arithmeticae. Gerh. Fleischer, Lipsiae, 1801, xviii+668 pp. [2] K. Hensel, Die multiplikative Darstellung der algebraischen Zahlen für den Bereich eines beliebigen Primteilers. J. f. d. reine und angewandte Math., 146 (1916), pp. 189-215.
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