What is meant by field, rings in maths
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Answer :
Field : A non empty set F equipped with two binary operations called addition and multiplication denoted by ( + ) and ( • ) is said to be a field if the following properties holds :
1) F is closed with respect to addition .
ie. a + b ∈ F for every a , b ∈ F .
2) Addition is associative .
ie. a + (b + c) = (a + b) + c for every a , b , c ∈ F .
3) Addition is commutative .
ie. a + b = b + a for every a , b ∈ F .
4) Existence of additive identity :
ie. there exists an additive identity in F denoted by 0 such that , 0 + a = a + 0 for every a ∈ F .
5) Existence of additive inverse :
ie. to each element a ∈ F , there exists an element -a ∈ F such that , -a + a = a + (-a) = 0 .
6) F is closed with respect to multiplication .
ie. a•b ∈ F for every a , b ∈ F .
7) Multiplication is associative .
ie. a•(b•c) = (a•b)•c for every a , b , c ∈ F .
8) Multiplication is commutative .
ie. a•b = b•a for every a , b ∈ F .
9) Existence of multiplicative identity .
ie. to each element a ∈ F , there exists 1 ∈ F such that , a•1 = 1•a = a .
10) Existence of multiplicative inverse .
ie. to each a (≠ 0) ∈ F , there exists an element a⁻¹ ∈ F such that , a•a⁻¹ = a⁻¹•a = 1 , where a⁻¹ = 1/a is the multiplicative inverse of a .
10) Multiplication is distributive under addition .
ie. a•(b + c) = a•b + a•c and (b + c)•a = b•a + c•a for every a , b , c ∈ F .
In other words , an algebraic structure (F,+,•) is said to be a field if ;
- (F,+) is an abelian group .
- (F-{0},•) is an abelian group .
- (F,+,•) holds distribute law .
- a•(b + c) = a•b + a•c
- (b + c)•a = b•a + c•a