Proof on Rational Numbers I am trying to determine whether the following structure forms a Ring under the Real Number Definition of Addition and Multiplication
Consider the set of Real Numbers of the form:
$A = \{a + bp \:|\: a,b \in \mathbb{Q}, p \in \mathbb{R} - \mathbb{Q}\}$ 
with $p$ being fixed. 
And consider the structure $\left(A, +, .\right)$
Where + and . are as defined in the Real Numbers. 
It's fairly easy to show that axioms over addition are satisfied and that it forms an Albelian Group. 
For multiplication however I don't believe that it's closed. 
Taking two elements in $A$ say 
$x_{1} = a_{1} + b_{1}p$ and $x_{2} = a_{2} + b_{2}p$ 
Then 
$x_{1} \cdot x_{2} = \left(a_{1}a_{2} + b_{1}b_{2}p^{2}\right) + \left(a_{1}b_{2} + a_{2}b_{1}\right)p$
Now unless I'm interpreting this incorrectly I can only say that it is of the form of A : $x = a + bp \iff p^{2} \in \mathbb{Q}$ 
If $p^{2} \in \mathbb{R} - \mathbb{Q}$ 
Then $x$ now takes the form 
$x = a + bp_{1} + cp_{2}$ where $a,b,c \in \mathbb{Q}$ and $p_{1}, p_{2} \in \mathbb{R} - \mathbb{Q}$ and $p_{1} \neq p_{2}$
As such I can not conclude that the set A is closed under multiplication unless $p^{2} \in \mathbb{Q}$ 
Have I missed something here?
Thanks, David 

You haven't missed anything. To be closed under multiplication $p$ should necessarily  be a root of a quadratic equation with rational coefficients. (It is called an algebraic number of degree 2).
