Question

Define field, Abelian group and ordered field in Maths for class 8

Answer 1

$\textbf{Field}$:

A non empty set is called field if it satisfies the following properties :

1. the cities in albanian group with respect to addition.

2. The non zero elements of the set full form in Abelian group with respect to multiplication.

3. Multiplication is to be distributed over addition.

$\textbf{Ordered Field }$:

Set is called an ordered field if it holds the following properties in addition to above Stated properties of field :

1. For any two rational numbers A and B only one of the following relations hold good
(I) a < b
(II) a > b
(III) a = b

2. If a, b, and c are three rational number and if a>b ,b >c then a>c .

3. If a, b, and c are three rational numbers a>b and c≠0, then a+c> b+c.

4. If a, b and c are three rational numbers and if a>b and c is a positive numbers then ac> bc.

$\textbf{Abelian Group }$:

If a group satisfy the commutative property then it is called an abelian group or commutative group.