Show that the closure of $\Bbb Q\cup\{i\}$ is a field

Question

Let K be the closure of $\Bbb Q\cup\{i\}$, that is, $K$ is the set of all numbers that can be obtained by (repeatedly) adding and multiplying rational numbers and $i$, where is the complex square root of -$1$. Show that $K$ is a field.

How about you start by listing the axioms of a field? This should be quite straightforward after that, and if you get stuck at any axiom you can edit you question for more explanation. — Ryan Sullivant, Sep 18 '13 at 17:38

score 2 · Answer 1 · answered Sep 18 '13 at 17:39

2

Hint: Show that $K=\{\,a+ib\mid a,b\in\mathbb Q\,\}$ and observe that $(a+bi)(a-bi)\in\mathbb Q$.

answered Sep 18 '13 at 17:39

Hagen von Eitzen

382,479

Thank you! This is very helpful! – Hao Sep 18 '13 at 17:42
See also http://math.stackexchange.com/questions/106330/field-extension-obtained-by-adjoining-a-cubic-root-to-the-rationals. – Dietrich Burde Sep 18 '13 at 18:06

Show that the closure of $\Bbb Q\cup\{i\}$ is a field

1 Answers1