Product of Units of Integral Domain with Finite Number of Units

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Theorem

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $D$ have a finite number of units.

Let $U_D$ be the set of units of $\struct {D, +, \circ}$.


Then:

$\displaystyle \prod_{x \mathop \in U_D} x = -1_D$


Proof

Consider the set $S$ defined as:

$S = U_R \setminus \set {1_D, -1_D}$

If $S$ has even cardinality, it can be partitioned into doubletons of the form $\set {u, u^{-1} }$.

Each of these doubletons has a product of $1_D$.

The product of all these with $1_D$ and $-1_D$ is $-1_D$.


It remains to be shown that $S$ cannot be of odd cardinality.

Aiming for a contradiction, suppose $S$ has an odd number of elements.

Then after pairing off each $u \in D$ with its product inverse $u^{-1}$, we are left with one element of $S$ which is self-inverse.

But from Self-Inverse Element of Integral Domain is Unity or its Negative, this self-inverse element is either $1_D$ or $-1_D$.

These have already been counted.

Hence there cannot be an odd number of elements of $S$.

$\blacksquare$


Sources