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Q2

What are the field axioms?in what respect does the field of real numbers differ from that of complex numbers?

Solution:

Field: A nonempty set F is said to be a field if for all x,y,z ∈F, the following axioms are satisfied:

1. x+y∈F

2. x+(y+z)=(x+y)+z

3. there exists 0 ∈ F such that x+0=x=0+x

4. There exists -x∈F such that x+(-x)=0=(-x)+x

5. x+y=y+x

6. x,y ∈F

7. x,(y,z)=(x,y,z

8. There exists 1 ∈F such that x.1=x=1.x

9. For x∈F-{0}, there exists x^{-1}∈ F such that x.x^{-1}=1=x^{-1}.x

10. x.y=y.x

11. x.(y+z)=x.y+x.z and (y+z).x=y.x+z.x

A field Fis a commutative ring in which every non-zero element has a multiplicative inverse. Thus (F,+) is a commutative group and if 0 denotes its zero element then (F\{0},� ) is a multiplicative group and the multiplication operation is distributive over the addition operation. In short, the addition and multiplication operations in a field F satisfy the ordinary rules of arithmetic, postulated as follows: For any a, b, c ÎF,

A field F is said to be finite if the set F has a finite number of elements.