Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$ So elements of this ring look like
$$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$
If I want to find the multiplicative inverse element for the above general element, then I'm trying to find $x,y,z\in\mathbb{Q}$ such that 
$$(a+b\sqrt[3]{2}+c\sqrt[3]{4})(x+y\sqrt[3]{2}+z\sqrt[3]{4})=1$$
I can see that expanding gives me the system 
$$ax+2cy+2bz=1$$
$$bx+ay+2cz=0$$
$$cx+by+az=0$$
I don't want to solve this using matrices because I know it will turn ugly.  Is there a more elegant way to approach the inverse calculation to avoid the ugly calculation?  The only thing I thought of was setting the bottom two equations equal to each other
$$bx+ay+2cz=cx+by+az$$
Which seems to indicate that 
$$a=b, b=c, a=2c$$ but this would make me think $a=b=c=0$ and thus a multiplicative inverse does not exist.
First let me remark that there is a general abstract argument which shows that if $L/K$ is a field extension and $a \in L$ is algebraic over $K$, then the $K$-algebra $K[a]$ is a field. Namely, $K[a]$ is an integral domain  which is also a finite-dimensional vector space over $K$. This implies for $0 \neq x \in K[a]$ that the linear map $K[a] \to K[a]$, $y \mapsto x \cdot y$ is surjective, since it is injective, which means that $x$ is invertible. To get a constructive proof, we just have to sit down and make the linear algebra argument here explicit.

The underlying vector space of $\mathbb{Q}[\sqrt[3]{2}]$ has basis $1,\sqrt[3]{2},\sqrt[3]{4}$. For some fixed non-zero element $x=a + b \sqrt[3]{2} + c \sqrt[3]{4}$, let's write down the linear map $y \mapsto x \cdot y$ in terms of this basis:

$~~~\,1 \mapsto a + b \sqrt[3]{2} + c \sqrt[3]{4}$

$\sqrt[3]{2} \mapsto 2c + a \sqrt[3]{2} + b \sqrt[3]{4} $

$\sqrt[3]{4} \mapsto 2b+2c\sqrt[3]{2} + a \sqrt[3]{4} $

The corresponding matrix is:

$$\begin{pmatrix} a &amp; 2c &amp; 2b \\ b &amp; a &amp; 2c \\ c &amp; b &amp; a \end{pmatrix}$$

From linear algebra we know how to invert matrices, for example via Cramer's rule. In this case, we get:

$$\frac{1}{a^3 - 6abc + 2b^3 + 4c^3} \cdot \begin{pmatrix} a^2 - 2bc &amp;  2b^2 - 2ac &amp;  4c^2 - 2ab \\  2c^2 - ab &amp;  a^2 - 2bc &amp;  2b^2 - 2ac \\ b^2 - ac &amp;  2c^2 - ab &amp;  a^2 - 2bc \end{pmatrix}$$

The determinant $a^3 - 6abc + 2b^3 + 4c^3$ has been computed via the Rule of Sarrus and  the cofactors have been computed by the usual formula for $2 \times 2$-determinants.

This matrix represents the linear map $y \mapsto x^{-1} \cdot y$ with respect to our basis. Thus, to get $x^{-1}$, we just have to evaluate at $1$, and we get:

$$x^{-1} = \frac{1}{a^3 - 6abc + 2b^3 + 4c^3} \cdot ((a^2 - 2bc) + (2c^2 - ab) \sqrt[3]{2} + (b^2 - ac) \sqrt[3]{4})$$

Of course, this method works quite generally. For example, for $x=a + b \sqrt[3]{p} + c \sqrt[3]{p}^2$ we have:

$$x^{-1} = \frac{1}{a^3 - 3pabc + pb^3 + p^2 c^3} \cdot ((a^2 - pbc) + (pc^2 - ab) \sqrt[3]{p} + (b^2 - ac) \sqrt[3]{p}^2)$$
Try these:
Rationalizing the denominator 3
Deradicalization of denominators