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I'm new to Field Theory and I'm looking for an example of an infinite simple extension.

Theorem: The element $\alpha$ is algebraic over $F$ if and only if the simple extension $F(\alpha)/F$ is finite.

Using the above theorem, I guess that something like $\Bbb{Q}({\pi})/\Bbb{Q}$ is an example of such an extension. But the proof of $\pi$ is transcendental is not at all trivial(and I don't think I can understand the proof with my limited knowledge).

So I have got two questions:

  • Is my example correct?

  • Are there any methods to construct an infinite simple extension which doesn't require more sophisticated tools?

Also, I would love to see some "trivial" examples(if they exist).

Any help is appreciated. Thank you.

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    $\begingroup$ Your example is correct. Any transcendental should work, and they should all look the same. You can just let $\alpha$ be any transcendental. $\endgroup$ Commented Jan 11, 2019 at 7:57

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Yes your example is correct. Another example would be $k(X)/k$ where $k(X)$ is the field of fractions of the polynomial ring $k[X]$, and $X$ is clearly transcendental.

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    $\begingroup$ An element is algebraic if it is killed by a polynomial, so by definition of $k[X]$ the element $X$ cannot be algebraic. $\endgroup$ Commented Jan 11, 2019 at 8:47

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