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How to prove or to disprove that the field of rational functions $\mathbb{Q}(x)$ over the rationals contains two subfields $K$ and $F$ such that $[\mathbb{Q}(x):K] < \infty$ and $[\mathbb{Q}(x):F] < \infty$ but $[\mathbb{Q}(x):K \cap F] = \infty$?

user64494
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    How do you know this happens? – Mariano Suárez-Álvarez Jul 05 '13 at 06:22
  • @ Mariano Suarez-Alvarez : Thank you. It has been fixed. – user64494 Jul 05 '13 at 06:34
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    Would this help? – Jyrki Lahtonen Jul 05 '13 at 06:44
  • @ Jyrki Lahtonen: Thank you for the interest to the question. It should be a more or less simple answer because the question was asked at one of student mathematical olimpiads in Ukraine. – user64494 Jul 05 '13 at 06:56
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    Olympiad kids (and others) can fire simple questions that may not have a simple answer. Anyway, may be you can use the answer by a sadly gone user? Does not use Galois theory AFAICT. – Jyrki Lahtonen Jul 05 '13 at 07:01
  • @ Jyrki Lahtonen: Indeed, your cited answer also answers the question under consideration. In fact, your answer uses authomorphisms. AFAIUI, that belongs to Galois theory. However, I accept the one as the answer to my question. – user64494 Jul 05 '13 at 07:11
  • If it is "students" mathematics Olim. then it probably contains stuff at undergraduate level, of which Galois Theory undoubtedly is part of. If it was a high school mathematics olympics...wow, are you kidding me? This would be the first time in my life I've ever seen such a question at this level for high school kids! – DonAntonio Jul 05 '13 at 08:18
  • @ DonAntonio : the question is not from a school mathematical olimpiad. – user64494 Jul 05 '13 at 08:26

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