Algebra - vocabulary

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F

field

noun

Pronunciation:

/fiːld/

Definition:

Field is an algebraic structure defined as 7-tuple of set, binary operations on this set (addition and multiplication), two unary operations (multiplicative inverse) and two nullary operations (0 and 1). With folowing axioms:

associativity of addition and multiplication
commutativity of addition and multiplication
0 and 1 are additive and multiplicative identity, it means that a + 0 = a and a· 1 = a for all a from field
for additive inverse (-a) holds: a + (−a) = 0
for multiplicative inverse (a^(-1)) holds: a· a⁻¹ = 1
distributivity of multiplication over addition: a· (b + c) = (a· b) + (a· c)

source:

https://www.macmillandictionary.com

https://en.wikipedia.org

Field of fractions

noun

pronunciation:

[ fēld əv ˈfrakSHəns ]

synonyms:

fraction field, field of quotients, or quotient field

meaning:

Field of fractions of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain $R$ are equivalence classes (see the construction below) written as ${\frac {a}{b}}$ with $a$ and $b$ in $R$ and $b\neq 0$ . The field of fractions of $R$ is sometimes denoted by $\operatorname {Frac} (R)$ or $\operatorname {Quot} (R)$ .

examples:

The field of fractions of the ring of integers is the field of rationals, i.e. $\mathbb {Q} =\operatorname {Frac} (\mathbb {Z} )$

Given a field $K$ , the field of fractions of the polynomial ring in one indeterminate $K[X]$ (which is an integral domain), is called the field of rational functions or field of rational fractions and is denoted $K(X)$ .