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Field Theory - General Question

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
I am studying field theory.

A general question I have is the following:

Let [TEX] E \supseteq F [/TEX] be fields and let [TEX] u \in E [/TEX].

Now, if I determine an irreducible polynomial f in F[x] such that f(u) = 0 in E, can I conclude that I have found the minimal polynomial of u over F.

Can someone please help?

Peter

[Note: This has also been posted on MHF]
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Recall that in a UFD, irreducibles are only unique "up to a unit factor" (and the units of a polynomial ring over a field are the non-zero field elements), so to conclude $f$ is THE minimal polynomial of $u$, we must have that $f$ is MONIC. For example:

$f(x) = 3x^2 - 6$ is A(n irreducible) polynomial in $\Bbb Q[x]$ for which we have $f(\sqrt{2}) = 0$, but it is not THE minimal polynomial of $\sqrt{2}$ precisely because it is not monic. To fix this, we have to multiply by the unit $\frac{1}{3}$.

Also, as a practical matter, determining that $f(u) = 0$ is usually the easy part. Proving irreducibility is often much more difficult, especially with polynomials of degree 4 or higher, which is why they are not often used as examples.