Field extensions

The question is: "Find a splitting field of

t^3-2

over

\mathbb{Q}

, and find the degree of that field over Q". Just wanted a working-check.

The polynomial splits over

Q(2^{3/2}, \mu)

where

\mu = \exp(\frac{2 \pi i}{3})

. Indeed, it has as its three roots

2^{3/2} \mu^i

for i=0,1,2. Clearly the cube root of 2 is not a rational multiple of mu, so the splitting field is generated by mu and 2^(3/2).

The degree of this field as an extension of Q: we use the tower law.

[\mathbb{Q}(2^{3/2}, \mu) : \mathbb{Q}(2^{3/2})]

is 3, since

t^3-1

is the min-poly of mu in Q(2^{3/2}).

[\mathbb{Q}(2^{3/2}):\mathbb{Q}]

is 3, since the minpoly of 2^{3/2} over Q is t^3-2.
Hence the degree is 9.

Is that correct? It feels a bit fluffy to me - seems a bit suspicious that so many of the degrees are 3.

Thanks!

Reply 1

DFranklin

18

Um, t^3-1 isn'[t minimal - it factorizes as (t-1)(t^2+t+1)

Reply 2

Smaug123

OP

15

Original post by DFranklin

Um, t^3-1 isn'[t minimal - it factorizes as (t-1)(t^2+t+1)

Oops - that was silly. I knew something was wrong, because I went through exactly the same thought processes at each step (whereas I'd have expected somewhere to have to think a little differently). I have also completely failed to manipulate indices correctly, and have concluded that the cube root of 2 is

2^{3/2}

.

So the answer is instead 6, because [Q(2^{1/3},mu): Q(2^{1/3})] is 2, while [Q(2^{1/3}):Q] is 3 because the minpoly of 2^{1/3} is t^3-2.

Thanks!

Reply 3

DFranklin

18

I agree 6 is the correct degree; I have to say I think I would have put in more justification (back in my exam days). E.g. you need to actually justify that t^2+t+1 doesn't factorize further in Q[2^{1/3}]. (It's actually quite easy here because Q[2^{1/3}] \subset R, so as soon as you show you need complex roots to factorize you're done). Similarly you need to justify that t^3-2 doesn't split in Q (again easy, because if it splits it has a linear factor, but you still have to do it IMHO).

I'd suggest you make sure you see at least one model answer for this type of question to get an idea about how much justification is expected.

(edited 9 years ago)

Reply 4

Smaug123

OP

15

Original post by DFranklin

I agree 6 is the correct degree; I have to say I think I would have put in more justification (back in my exam days). E.g. you need to actually justify that t^2+t+1 doesn't factorize further in Q[2^{1/3}]. (It's actually quite easy here because Q[2^{1/3}] \subset R, so as soon as you show you need complex roots to factorize you're done). Similarly you need to justify that t^3-2 doesn't split in Q (again easy, because if it splits it has a linear factor, but you still have to do it IMHO).

I'd suggest you make sure you see at least one model answer for this type of question to get an idea about how much justification is expected.

Will do (giving sufficient justification might have made me spot that t^3-1 wasn't irreducible, anyway).

Thank you!

Field extensions

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