Let Q be the field of all rational numbers, show that Q(i)={a+bi|a,b e Q} is a field if i is define at sqrt(-1) under the addition and multiplication of complex numbers.
This is a straightforward matter of checking the various field properties.
The only difficult one is multiplicative inverse of non-zero terms.
\(\displaystyle \frac{1}{{a + bi}} = \frac{a}{{a^2 + b^2 }} - \frac{b}{{a^2 + b^2 }}i\).
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