# Minimal polynomials and field extensions.

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#1
We have been given that [M:Q]=4 and has assumed that there exists a u in M such that M=Q(u) and the minimal polynomial of u is of the form
X^4+aX^2+b with a and b in Q. We are trying to show there exists a field L such that Q ⊂ L ⊂ M and [L:Q]=2.

I have decided that there is a polynomial Y^2+aY+b which is monic, irreducible with u^2 being a root. I believe this is the root to proving the above statement but not sure how to get there. I think this is the minimal polynomial of a field extension but not quite sure what this field extension is.

Thank you in advance for any help given.
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2 years ago
#2
(Original post by mollyjordansmith)
We have been given that [M:Q]=4 and has assumed that there exists a u in M such that M=Q(u) and the minimal polynomial of u is of the form
X^4+aX^2+b with a and b in Q. We are trying to show there exists a field L such that Q ⊂ L ⊂ M and [L:Q]=2.

I have decided that there is a polynomial Y^2+aY+b which is monic, irreducible with u^2 being a root. I believe this is the root to proving the above statement but not sure how to get there. I think this is the minimal polynomial of a field extension but not quite sure what this field extension is.

Thank you in advance for any help given.
Consider the roots of your polynomial Y.
0
X

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