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Proposition: Complex Numbers are a Field Extension of Real Numbers

Let \((\mathbb R,\oplus,\odot)\) be the field of real numbers and let \((\mathbb C, + ,\cdot)\) be the field of complex numbers. We define a function. \[f:=\cases{\mathbb R\to\mathbb C,\\ x\to (x,0).}\]

Then \(f\) is an bijective field homomorphism, i.e. for all \(a,b\in\mathbb R\), we have

\[\begin{array}{rcl} f(a\oplus b)&=&f(a) + f(b),\\ f(a\odot b)&=&f(a) \cdot f(b). \end{array}\]

The following statements are equivalent:

• "The field of real numbers is embedded in the field of complex numbers."
• "The field of complex numbers is a field extension of the field of real numbers".

Table of Contents

Proofs: 1

Mentioned in:

Chapters: 1
Definitions: 2 3
Proofs: 4 5 6

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References

Bibliography

1. Timmann, Steffen: "Repetitorium der Funktionentheorie", Binomi-Verlag, 2003