Question

Show that C is a field

Answer 1

Show that C is a field

It has to satisfy :The set of complex numbers C is a field with the binary operations of addition, +, and multiplication ,*

As C is a field with the operations + and * C must satisfies all of the field axioms with these operations. Let z=a+bi, w=c+di, v=e+fi?C. ( z,w,v are in x+iy form)

• Addition):z+w=(a+bi)+(c+di)=(a+c)+(b+d)i?C
• Associativity of Addition:

  z+(w+v)=[a+bi]+([c+di]+[e+fi])=[a+bi]+((c+e)+(d+f)i)=(a+(c+e))+(b+(d+f))i=((a+c)+e)+((b+d)+f)i=((a+c)+(b+d)i)+[e+fi]=([a+bi]+[c+di])+[e+fi]=(z+w)+v

Commutativity of Addition:

z+w=(a+bi)+(c+di)=(a+c)+(b+d)i=(c+a)+(d+b)i=(c+di)+(a+bi)=z+w

Multiplication rule:

i) Multiplication

zw=(ac?bd)+(ad+bc)i?C

ii)Commutativity of Multiplication

zw=(ac?bd)+(ad+bc)i=(ca?db)+(da+cb)i=wz

iii)Associativity of Multiplication):

z(wv)=z[(ce?df)+(cf+de)i]=[(ace)?(adf)]?[(bcf)?(bde)]+[((acf)+(ade))+(bce)?(bdf)]i

zw)v=[(ac?bd)+(ad+bc)i]v=[(ace)?(bde)]?[(adf)+(bcf)]+[((acf)?(bdf))+((ade)+(bce))]i

Iv) if we multiply with 1 , a+bi)?1=a+bi=z will come

C satisfies all of the Rules with the operations + and *. Hence C is a field.

Note: other sub rules can be varified, like identity, inverse. but basic rules are in above and proved.