Source Word
field of quotients
Noun
(algebra)
A
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')=(ab'+a'b,bb'),
the
multiplicative
operator
is
defined
coordinate-wise,
the
zero
is
(0,1),
the
unity
is
(1,1),
the
additive
inverse
of
(a,b)
is
(-a,b),
equivalence
is
defined
like
so:
(a,b)≡(a',b')
if
and
only
if
ab'=a'b,
and
multiplicative
inverse
of
a
non-zero–equivalent
element
(a,b)
is
(b,a).
Japanese Meaning
「field of quotients」とは、与えられた整域に対して、その各元を整域の非零元による順序対として表現し、加算および乗算が定義される体のことであり、分数体とも呼ばれます。
Sense（1）
(algebra)
A
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')=(ab'+a'b,bb'),
the
multiplicative
operator
is
defined
coordinate-wise,
the
zero
is
(0,1),
the
unity
is
(1,1),
the
additive
inverse
of
(a,b)
is
(-a,b),
equivalence
is
defined
like
so:
(a,b)≡(a',b')
if
and
only
if
ab'=a'b,
and
multiplicative
inverse
of
a
non-zero–equivalent
element
(a,b)
is
(b,a).
（ plural ）
