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John Wilson (1741-1793) was a well-known English mathematician in his time, whose legacy lives on in his eponymous result, *Wilson’s Theorem*. To recall, this is the statement that an integer is prime if and only if

(The “if” part is trivial.) As is the case for many historical results, Wilson’s Theorem was *not *proven by Wilson. Instead, it was Joseph Lagrange who provided the first proof.

The proof, as we see it today, might be phrased as follows:

*Proof: *Suppose that is prime. Then each of the nonzero residues modulo is a unit, so that represents the product over all units in . If

, i.e. ,

then and its inverse each show up in our list of units. We cancel out such terms in pairs, and conclude that

We have if and only if , which by primality of forces or . In other words, . It follows that

When is composite, the direct translation of Wilson’s problem gives

The problem, here, is that we’ve multiplied a number of zero divisors together, which can be avoided by only multiplying across the units, , of . In this post, we consider the product

determine its value, give credit to Gauss for doing so over two centuries ago, and discuss a few generalizations.

Let be a number field, i.e. a finite field extension to . We recall that the * ring of integers* in

*k*,

*denoted , is the ring*

For , the ring of integers is just the integers , in which case we recall the Fundamental Theorem of Arithmetic: that every integer may be written as a finite product

in which the are prime and uniquely determined (up to permutation). Domains for which this holds are known in general as ** unique factorization domains **(UFDs). For — with square-free — the ring of integers will in general

*not*be a UFD. In fact, for , the integers have unique factorization only in the 9 cases

Far less is known in the case (in which case *k* is known as a * real quadratic field*), although an unproven conjecture dating back to Gauss suggests that there should be infinitely many real quadratic fields. More recently, some heuristics stemming from Cohen suggests that the ring of integers in should be a UFD with probability as on the square-free integers.

Here, we’ll focus on a more tractable variant of this problem:

**Question:** What can be said about the number of distinct real quadratic fields with for which is ** not** a UFD?

For a weak answer to the question above, we devote the rest of this article to the establishment of the following bound:

**Theorem: **As , we have

in which the implied constant is made effective (e.g. greater than ).

For what follows, we define a * loaded die *as a discrete probability distribution with six outcomes (labelled {1,…,6}), each of which has positive probability. To each such die, we associate a generating polynomial, given by

in which denotes the probability of the outcome *i*. If corresponds to another such die, we note that the product

has coefficients which reflect the probabilities of certain dice sums for *p* and *q *(and this is the utility of generating polynomials). We are now ready to ask the following question:

**Question: ***Does there exist a pair of loaded dice such that the probability of rolling any dice sum ({2,…12}) is equally likely?*