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