I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory textbook, the Norm of a quaternion $\alpha=a+bi+cj+dk$ is $N\alpha=a^2+b^2+c^2+d^2$ and the trace of $\alpha$ is $Tr(\alpha)=2a$. I've been trying to relate this to the definition given by another book on field theory, which says we can create a linear map that represents multiplication by a quaternion element such as the matrix
$$m = \left(\begin{matrix} a & b &c&d\\ -b&a&-d&c\\-c&d&a&-b\\-d&-c&b&a \end{matrix}\right)$$
and that if $m:V\rightarrow V$ is a linear operator on a finite dimentional vector space V over F, the trace of $m$ is the trace of the matrix $m$ and the norm is the norm of the matrix $m$. However, the trace and norm of this matrix are inconsistent with the first definition. (The trace of the matrix is $4a$ and the norm is the $(a^2+b^2+c^2+d^2)^2$. Is one of my definitions incorrect, or is there something more to it?
Your computations are correct. In particular, the norm form of an algebra of dimension $r$ over a field $K$ is always of degree $r$.
However, as you noted, in the case of quaternions, the norm form is the square of the quadratic form of signature $(4,0)$ which can also be obtained as $$ N(q)=q\overline q $$ where $\overline q$ denotes quaternionic conjugation.
Then one more properly defines reduced trace and norm the quantities $$ t(q)=q+\overline q\qquad N(q)=q\overline q $$ and observe that a quaternion $q$ always satisfies the identity $$ q^2-t(q)q+N(q)=0. $$ In fact the reduced trace and norm of $q$ are the complex trace and norm of $q$ under the identification $\Bbb R(q)\simeq\Bbb C$.