SOLUTION
The set of complex numbers, C with addition and multiplication is defined as a field comprising of additive and multiplicative credentials (0,0) and (1,0).
With the complex number i=(0,1), it is possible to interchangeably list each complex number as
z=(x,y)=x+(i×y)=x+iy
Now, in order to prove that set of complex numbers is field, we need to find the field axioms for the definitions of addition and multiplication i.e.
z+w=(x,y)+(u,v)=(x+u,y+v)
z×w=(x,y)×(u,v)=(x×u−y×v,x×v+y×u)
Now, for the first case; (when (+) and (×) are associative). It is simple for addition, but multiplication requires illustrations that actually a×(b×c)=(a×b)×c
Now, let a=(x1,y1),b=(x2,y2),and,c=(x3,y3) such that;
a.(b.c)=(x1,y1).((x2,y2).(x3,y3)) =(x1,y1).(x2x3−y2y3,x2x3+y2y3)
=(x1(x2x3−y2y3)−y1(x2x3+y2y3),x1(x2x3+y2y3)+y1(x2x3−y2y3))
=(x1x2x3−x1y2y3−y1x2x3−y1y2y3,x1x2x3+x1y2y3+y1x2x3−y1y2y3)
=(x3(x1x2−y1y2)−y3(x1y2+y1x2),x3(x1y2+y1x2)+y3(x1x2−y1y2))
=(x1x2−y1y2,x1y2+y1x2).(x3,y3)
=((x1,y1).(x2,y2)).(x3,y3)
=(a.b).c
For commutative of both (+) and (×), we can prove that
a+b=b+a and a×b=b×a
(x1+y1)+(x2+y2)=(x1+x2)+(y1+y2)
=(x2+x1)+(y2+y1)
=(x2+y2)+(x1+y1)
=b+a
This also applies to the multiplication, (×)
For the distributive law a×(b+c)=(a×b)+(a×c) we have;
a×(b+c)=(x1+y1)×([x2+x3]+[y2+y3])
=x1[x2+x3]+x1[y2+y3]+y1[x2+x3]+y1[y2+y3]
=[(x1x2−y1y2)+(x1y2+y1x2)]+[(x1x3−y1y3)+(x1y3+y1x3)]
=(a×b)+(a×c)
For the additive identity, (0,0) we have;
0=0+0i.
That is z1+0=(x1+y1i)+(0+0i)
=(x1+0)+(y1+0)i
=x1+y1i
=z1
For the multiplicative identity (1,0) we have
1=1+0i
That is z1×1=(x1+y1)(1+0i)
=(x1+x1×0i)+(y1+y1×0)i
=x1+y1i
=z1
For the additive inverse of (x,y) gives (−x,−y) i.e. Considering;
z1 which is −z1=−x−yi
That is z1+(−z1)=(x1+y1i)+(−x1−y1i)
=(x1−x1)+(y1−y1)i
=0+0i
=0
For the multiplicative inverse to (x,y) is ((x2+y2)x,−(x2+y2)y) i.e.
considering complex number z1 where z1=/0
z1−1=x12+y12x1−y1i
Now z1×z1−1=(x1+y1i)×(x12+y12x1−y1i)
=x12+y12(x1+y1i)(x1−y1i)
=x12+y12x12−x1y1i+x1y1i−y1i2
=x12+y12x12+y12
=1
Therefore, Since the set of complex numbers C used in the proof above
satisfies the axioms working with the addition and multiplication, it can therefore be concluded that C is a field.
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