Definitions
1.NounA field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so:
(
a
,
b
)
+
(
a
′
,
b
′
)
=
(
a
b
′
+
a
′
b
,
b
b
′
)
{\displaystyle (a,b)+(a',b')=(ab'+a'b,bb')}
, the multiplicative operator is defined coordinate-wise, the zero is
(
0
,
1
)
{\displaystyle (0,1)}
, the unity is
(
1
,
1
)
{\displaystyle (1,1)}
, the additive inverse of
(
a
,
b
)
{\displaystyle (a,b)}
is
(
−
a
,
b
)
{\displaystyle (-a,b)}
, equivalence is defined like so:
(
a
,
b
)
≡
(
a
′
,
b
′
)
{\displaystyle (a,b)\equiv (a',b')}
if and only if
a
b
′
=
a
′
b
{\displaystyle ab'=a'b}
, and multiplicative inverse of a non-zero–equivalent element
(
a
,
b
)
{\displaystyle (a,b)}
is
(
b
,
a
)
{\displaystyle (b,a)}
.