Is $\mathbb R^2$ a field? I'm new to this very interesting world of mathematics, and I'm trying to learn some linear algebra from Khan academy.
In the world of vector spaces and fields, I keep coming across the definition of $\mathbb R^2$ as a vector space ontop of the field $\mathbb R$. 
This makes me think, Why can't $\mathbb R^2$ be a field of its own? 
Would that make $\mathbb R^2$ a field and a vector space?
Thanks

Adding to the above answer. With the usual exterior multiplication of the $\mathbb{R}-\{0\}$ as a ring with the natural addition and multiplication you can not make a field out of $\mathbb{R}^{2}-(0,0)$
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But there may exist other products such as the one in the answers which can make a field out of ${\mathbb{R}\times\mathbb{R}}-\{ 0\}$
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According to one of the theorems of Field theory every field is an Integral domain. So by considering :
${\mathbb{R}\times\mathbb{R}}-\{ 0\}$
With the following natural product:
$(A,B)*(C,D)=(AB,CD) $
We see that 
$(1,0)*(0,1)=(0,0) $
Which means that $\mathbb{R}^{2} $is not an integral domain and hence not a field.
