Complex numbers/Field/Fact/Proof

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The field properties for the addition are clear, since the corresponding properties hold for . We have

so is the neutral element for the multiplication. The commutativity of the multiplication follows directly from its formula. To show associativity of the multiplication we compute

We also get

Suppose now that

Then at least one of the numbers or is different from and therefore . Hence is a complex number and

so every element has an inverse with respect to the multiplication. The distributivity law follows from