(New - Updated 3/1/13) Is this argument correct in showing that this field extension does not have the stated degree?

Question

I have been asked to prove that $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}]=4$. I have some problems believing this to be true and have the following argument that assumes it to be true and seemingly derives a contradiction. If someone could verify whether in the following argument I have made any incorrect assertions or otherwise, I would be grateful.

1. By the Tower Law for field extensions, we have: $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}]= [\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}(\sqrt2)]\cdot[\mathbb{Q}(\sqrt2):\mathbb{Q}]$. We have that $[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2$. I have previously shown this result explicitly and have no problem working with this easy vector space over $\mathbb{Q}$.
2. So, if the question is correct, then $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}(\sqrt2)]=2$. For a simple field extension $K(\alpha)$ over $K$, with $\alpha$ algebraic over $K$, we have $[K(\alpha):K]=$ deg(mipo$_K(\alpha))$. We need to show $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}(\sqrt2)]=[\mathbb{Q}(\sqrt2)(\sqrt{1+i}):\mathbb{Q}(\sqrt2)]=2$. Therefore, by the result just stated, with $\mathbb{Q}(\sqrt2)$ as our base field and $\sqrt{1+i}$ as the algebraic element we adjoin, we have that deg(mipo$_{\mathbb{Q}\sqrt2}(\sqrt{1+i}))=2$.
3. If this is the case, then we have that for some $p,q\in \mathbb{Q}(\sqrt2)$ that $p+q\sqrt{1+i}+(1+i)=0$. But there are no non-trivial solutions to this equation, since writing $\sqrt{1+i}$ explicitly, i.e. $\sqrt{1+i}=\frac{\sqrt{1+\sqrt2}}{\sqrt2}+i\frac{\sqrt{-1+\sqrt2}}{\sqrt2}$ (I derived this form in radicals and it can be verified on Wolfram), we have that this equation is equivalent to $(a+b\sqrt2)+(c+d\sqrt2)(\frac{\sqrt{1+\sqrt2}}{\sqrt2}+i\frac{\sqrt{-1+\sqrt2}}{\sqrt2})+(1+i)=0$, where $a,b,c,d,e,f\in\mathbb{Q}$.
4. Since the equation equals $0$, in particular the imaginary component vanishes, i.e. $(c+d\sqrt2)(\frac{\sqrt{-1+\sqrt2}}{\sqrt2})+1=0$. Ultimately, no consistent solution exists to this equation.
5. Rearranging we obtain, $\sqrt{-1+\sqrt2}=\frac{-\sqrt2}{c+d\sqrt2}$, and after some rationalising it follows that $\sqrt{-1+\sqrt2}=\frac{2d}{c^{2}-2d^{2}}-\frac{c}{c^{2}-2d^{2}}\sqrt2$.
6. Hence, $\sqrt{-1+\sqrt2}=p+q\sqrt2$, for $p,q \in \mathbb{Q}$ as in the previous step. But, by first squaring both sides, a contradiction is derived. After squaring, we have the quadratic equation: $-1+\sqrt2=p^{2}+2q^{2}+2pq\sqrt2$
7. From this, we obtain two equations, which are: $p^{2}+2q^{2}=-1$ and $2pq=1$. Solving does not lead to any solutions in $\mathbb{Q}$ as, for instance $q^2=\frac{-4\pm\sqrt{-16}}{16}$, contradicting $p,q\in\mathbb{Q}$. Indeed, $p^{2}+2q^{2}\neq-1$ in $\mathbb{Q}$.
8. Therefore, by our chain of reasoning, deg(mipo$_{\mathbb{Q}\sqrt2}(\sqrt{1+i}))\neq2$. Hence by the earlier result, $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}(\sqrt2)]\neq2$ and thus by the first point (using the Tower Law), $[\mathbb{Q}(\sqrt2,\sqrt{1+i}):\mathbb{Q}]\neq4$.

Please note that I have updated this solution after the feedback. Thank you all for your initial feedback (particularly T. Bongers). However, after writing the minimal polynomial in degree 2 (initially I had written it in degree 1), I believe that a contradiction also arises. I would appreciate any feedback as to whether the updated reasoning here is sound.

Answer 1

Your step three isn't correct. It should read

There exist $p, q \in \mathbb{Q}(\sqrt{2})$ with $$\sqrt{1 + i}^2 + p \sqrt{1 + i} + q = 0$$

Your step uses a minimum polynomial of degree less than $2$.