# Field extension if and only if proof

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#1
Let F be an extension of Q with [F : Q] = 4. Prove that there exists a ﬁeld L such that Q ⊂ L ⊂ F and [L : Q] = 2 if and only if there exists u ∈ F such that F = Q(u) and the minimal polynomial of u is of the form X^4 + aX^2 + b with a,b ∈Q
Q is the quotient ring.
I have started to do the first implies but not sure how I am meant to use [L:Q]=2 to help me show F=Q(u) with the specific minimal polynomial. I have used the Tower Law to split the [F:Q]=[F:L][L:Q] to find [F:L]=2 but not sure if that is helpful at all.
For the second implies I do not know where to start.

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2 years ago
#2
(Original post by mollyjordansmith)
Let F be an extension of Q with [F : Q] = 4. Prove that there exists a ﬁeld L such that Q ⊂ L ⊂ F and [L : Q] = 2 if and only if there exists u ∈ F such that F = Q(u) and the minimal polynomial of u is of the form X^4 + aX^2 + b with a,b ∈Q
Q is the quotient ring.
I have started to do the first implies but not sure how I am meant to use [L:Q]=2 to help me show F=Q(u) with the specific minimal polynomial. I have used the Tower Law to split the [F:Q]=[F:L][L:Q] to find [F:L]=2 but not sure if that is helpful at all.
For the second implies I do not know where to start.

I would start by showing that if H is an extension of K with [H:K]=2, then we can find k in K and h in H with H=K(h) and h^2 = k. (More colloquially, we can get H by adding a square root of an element of K).
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#3
(Original post by DFranklin)
I would start by showing that if H is an extension of K with [H:K]=2, then we can find k in K and h in H with H=K(h) and h^2 = k. (More colloquially, we can get H by adding a square root of an element of K).
Thank you so much.
From here I have deduced that:
If [M:L]=2 then there exists an a in M such that M=L(a) and a^2 is in L
If [L:Q]=2 then there exists a b in L such that L=Q(b) and b^2 is in Q
A basis of M over L is {1,a}
A basis of L over Q is {1,b}
A basis of M over Q is {1, a, b, ab}
so
a^2=(m+nb)^2=m^2+2mnb+n^2b^2 with m,n in Q

Apparently this is all I need to now show M=Q(u) for u in M and to find the minimal polynomial however I am struggling to string it all together and so if you have any hints or tips for that, that would be helpful. Thank you again.
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#4
(Original post by mollyjordansmith)
Let F be an extension of Q with [F : Q] = 4. Prove that there exists a ﬁeld L such that Q ⊂ L ⊂ F and [L : Q] = 2 if and only if there exists u ∈ F such that F = Q(u) and the minimal polynomial of u is of the form X^4 + aX^2 + b with a,b ∈Q
Q is the quotient ring.
I have started to do the first implies but not sure how I am meant to use [L:Q]=2 to help me show F=Q(u) with the specific minimal polynomial. I have used the Tower Law to split the [F:Q]=[F:L][L:Q] to find [F:L]=2 but not sure if that is helpful at all.
For the second implies I do not know where to start.

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 proved 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. All we know about u is that it is in M and is a root of
X^4+aX^2+b

Thank you in advance for any help given.
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2 years ago
#5
(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 proved 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. All we know about u is that it is in M and is a root of
X^4+aX^2+b

Thank you in advance for any help given.
To recap:

You know M=Q(u), and u^2 is a root of Y^2+aY+b.

Now, you are looking for an element of M you can adjoin to Q to get an inbetween field L. There's a fairly obvious element to try (you'll still need to show it works). Can you see any such element?
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