Proof: K is a Root Field for Every Irreducible Polynomial with a Root in K

[Kiwi1]

Suppose [K:F]=n, where K is a root field over F. Prove K is a root field over F of every irreducible polynomial of degree n in F[x] having a root in K.

I don't believe my solution to this problem because I 'prove' the stronger statement: "K is a root field over F for every irreducible polynomial in F[x] having a root in K."

Have I done something wrong?

I have
"Theorem 7: Let K be the root field of some polynomial q(x) over F. For every irreducible polynomial p(x) in F[x], if p(x) has one root in K, then p(x) must have all of its roots in K."

My solution:
K is a root field so there exists a polynomial p(x) such that K is the root field of p(x) over F.
Suppose there is another polynomial b(x) with a root in K and root field \(K_1\). By theorem 7 all of the roots of b(x) are in K. So \(K \supseteq K_1\).
Now \(K_1\) is a root field, what's more it contains a root of p(x) so by Theorem 7 \(K_1\) contains all of the roots of p(x).
Therefore \(K \subseteq K_1\)

and \(K = K_1\)

 

[jvicens]

so K is a root field over F for every irreducible polynomial in F[x] having a root in K.No, you have not done anything wrong. Your solution is correct and proves the stronger statement. In fact, your proof is a generalization of the original statement and is a stronger result. This means that your proof can be used to prove both the original statement and the stronger statement. This is a common occurrence in mathematics, where a more general result is proven that encompasses a specific case. So, your solution is valid and does not have any errors.