field of quotients

Meaning

Noun

• 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: $(a,b)\; +\; (a\text{'},b\text{'})\; =\; (a\; b\text{'}\; +\; a\text{'}\; b,b\; b\text{'})$, 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: $(a,b)\; \backslash equiv\; (a\text{'},\; b\text{'})$ if and only if $a\; b\text{'}\; =\; a\text{'}\; b$, and multiplicative inverse of a non-zero–equivalent element $(a,b)$ is $(b,a)$.

Related

Similar words

• field of fractions