### Fields and Integral Domains

We have been treating *L*_{f} as a
commutative ring. More restrictive categories from abstract algebra are
*fields* and *integral domains*. A field is essentially a
commutative ring with a multiplicative identity and reciprocals for all elements.
An integral domain is a commutative ring with a multiplicative identity and no
divisors of 0.

We know *L*_{f} has a multiplicative identity (namely 1).
We know *L*_{q} *L*_{f}
can have elements with no reciprocals and divisors of 0, but does this apply to
all number bases?
The following are properties that *L*^{k}
may or may not have for a given k.

**LP1:**
m such that 1/m

**LP2:**
m,n m0, n0
such that mn = 0.

**LP3:**
m such that a,b,c
abc such that
a^{2} = b^{2} = c^{2} = m.

**LP1** is the non-existence of all reciprocals.
**LP2** is the existence of divisors of 0.
**LP3** is the existence of more than two square roots for some number.

**LP Conjecture:**
LP1 LP2 LP3 LP1 and
~LP1 ~LP2 ~LP3 ~LP1

By the Number of Square Roots Conjecture,
this would suggest that *L*^{k} is a field and an integral
domain if k is prime or the power of a prime and isn't a field
or an integral domain otherwise.

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© Scott Sherman 1999