Why in most of exercise of Linear Algerbra field involved is a subfield of complex numbers I am studying Linear Algebra by Hoffman , they have written that reader should assume that field involved is a subfield of complex numbers , they have explained a reason beyond this by giving the argument of field of characterstic zero .can anyone explain regarding this ? Thanks
 A: One of the things we are used to doing in a field is division. For example, solving a quadratic equation $x^2 + px + q = 0$ involves dividing by $2$. However, if $1 + 1 = 0$, dividing by $2$ is not allowed, as $2 = 0$; it would be the same as saying $0$ has a multiplicative inverse, which can never happen in any ring where $0 \neq 1$.
Some concepts, like skew-symmetric matrices, do not mean the same thing in a field of characteristic $2$. Other concepts, like positive-definite (needed to define inner products) do not make sense unless one has an ordered subfield, which is never possible in a field of finite characteristic (you can define an order, but it doesn't "respect" the field operations).
It may seem strange that finite fields even exist, but they do, and in such a field (because they are finite):
$1\\
1+1\\
1+1+1$
et cetera, cannot ALL be different. If we denote $1 + 1 +\cdots + 1\ (k\ \text{times})$ as $k\cdot 1$, then if:
$k\cdot 1 = m\cdot 1$, for $k < m \in \Bbb Z^+$ we have:
$0 = k\cdot 1 - k\cdot 1 = m\cdot 1 - k\cdot 1 = (m-k)\cdot 1$, and $m - k$ is an example of such an $n$.
It turns out that when this happens, for us to actually get a field, the smallest such positive $n$ must be a prime number, $p$. Vector spaces over such finite fields are often used in "coding" information, and error correction.
When first undertaking a study of linear algebra, assuming $F$ is a subfield of the complex numbers allows us to perform our usual "arithmetic" without fear of getting unusual results. Generalizing what is learned to an arbitrary field is usually not much harder, but one has to pay attention to any proof where one divided by a scalar, as there might be exceptions in a field of finite characteristic (these will often be pointed out in the text).
