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This question was asked in my Field Theory Quiz and I Think I need help to prove it.

Let F be an algebraic extension of K such that every polynomial in K[x] has a root in F. Then show that F is an algebraic closure of K.

Attempt: F will be called Algebraically closed field if every polynomial $ f\in F[x]$ has a root in F. Let f be given polynomial with coefficients in F. Then I need too prove that f has at least 1 root in F using only that any polynomial in K[x] has a roots in F. I am at loss of ideas and unable to move foreward.

I removed proof verification tag as it was a terrible proof.

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    The proof isn’t correct, you need to show that every polynomial in $F$ has a root in $F$, which is stronger than every polynomial in $K$ having a root in $F$. – Andrew Dudzik Sep 21 '21 at 07:44
  • @Slade I am working on it again now as it was posted 8 months ago. I will add a new attempt –  Sep 21 '21 at 07:46
  • @Slade I think proof of user : cos_dm_math21 is great. –  Sep 21 '21 at 08:01
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    You question has inconsistencies. At one place (the quotation) you state that every polynomial in $K[x] $ has a root in $F$ and in your attempt you assume that every polynomial in $K[x] $ has all roots in $F$. The accepted answer also uses the same assumption. – Paramanand Singh Sep 21 '21 at 09:37
  • @ParamanandSingh Thanks for pointing out. I am editing and I am looking for solution with has any polynomial in K[x] has a root in F. Can you please guide? –  Sep 21 '21 at 10:41
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    There is a standard theorem by name of Gilmer which says that if every polynomial in $K[x] $ has a root in $F$ then every polynomial in $K[x] $ has all roots in $F$. Is it provided in your textbook? The proof is a bit complicated. – Paramanand Singh Sep 21 '21 at 12:04
  • @ParamanandSingh No unfortunately it's not. I followed Algebra by Thomas Hungerford and it was not part of text there. –  Sep 21 '21 at 12:05
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    I will check if this is available on mathse – Paramanand Singh Sep 21 '21 at 12:06
  • @ParamanandSingh Thank you very much! –  Sep 21 '21 at 12:06
  • See the references in this answer: https://math.stackexchange.com/a/3507595/72031 – Paramanand Singh Sep 21 '21 at 12:07
  • @ParamanandSingh I am a bit busy. Will check after some time. Thanks a lot for your guidence! –  Sep 21 '21 at 12:08

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