MHB Field Theory - Algebrais Extensions - D&F - Section 13.2 - Exercise 4, page 530

[Math Amateur]

Can someone help me get started on the following problem.

Determine the degree over \mathbb{Q} of \ 2 + \sqrt{3} and of 1 + \sqrt[3]{2} + \sqrt[3]{2}

Peter

[This has also been posted on MHF]

 

[Deveno]

Let's try a naive approach:

Let $u = 2 + \sqrt{3}$. Then $(u - 2)^2 = 3$, so:

$(u - 2)^2 - 3 = 0$.

Multiplying this out, we get:

$u^2 - 4u + 1 = 0$, that is, $u$ is a root of the quadratic polynomial:

$x^2 - 4x + 1$.

If one is willing to accept that $u \not\in \Bbb Q$, this of course means that:

$x^2 - 4x + 1$ is irreducible over $\Bbb Q$. Can you continue?

 

[Math Amateur]

Let's try a naive approach:

Let $u = 2 + \sqrt{3}$. Then $(u - 2)^2 = 3$, so:

$(u - 2)^2 - 3 = 0$.

Multiplying this out, we get:

$u^2 - 4u + 1 = 0$, that is, $u$ is a root of the quadratic polynomial:

$x^2 - 4x + 1$.

If one is willing to accept that $u \not\in \Bbb Q$, this of course means that:

$x^2 - 4x + 1$ is irreducible over $\Bbb Q$. Can you continue?

Thanks Deveno. Sorry for late reply - day job intervened in things!

I followed your post and assume that all we have to do now is quote Theorem 4 of Dummit and Foote section 13.1 which states the following:
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Theorem 4. Let $$ p(x) \in F[x] $$ be an irreducible polynomial of degree n over a field F and let K be the field $$ F[x]/(p(x))$$. Let $$ \theta = x mod(p(x)) \in K $$. Then the elements

$$ 1, \theta, {\theta}^2, ... ... , {\theta}^{n-1} $$

are a basis for K as a vector space over F, so the degree of the extension is n i.e.

$$ [K \ : \ F] = n $$. ... ... ... etc etc

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In our situation we have

$$ p(x) = x^2 - 4x - 2 $$ where $$ p(x) \in \mathbb{Q}[x] $$.

p(x) has a root u in $$ \mathbb{Q}[x]/(p(x)) $$.

Specifically $$ u = x \ mod(p(x)) \in K = \mathbb{Q}(2 + \sqrt{3}) $$

Then the elements 1 and u are a basis for $$ K = \mathbb{Q}(2 + \sqrt{3})$$.

Thus the degree of the extension $$ \mathbb{Q}(2 + \sqrt{3}) $$ over $$ \mathbb{Q} $$ is 2.

That is [K : F] = 2

Can someone please confirm that the above is correct?

Peter

 

[Deveno]

More or less.

You have to realize that $\Bbb Q[x]/(p(x))$ doesn't EQUAL $\Bbb Q(u)$, but they are ISOMORPHIC.

So you need one small theorem from Linear Algebra:

Isomorphic vector spaces over a common field have the same dimension.

This is what allows us to conclude that $[\Bbb Q(u):\Bbb Q] = \text{deg}(p)$.

It may be instructive to convince yourself that an isomorphism between two extension fields of a base field $F$ that preserves the base field is an $F$-linear map (this will be helpful to you later on when you consider automorphisms of an extension field).

Another approach:

Show that: $\Bbb Q(2 + \sqrt{3}) = \Bbb Q(\sqrt{3})$.