TITLE: Construct a field having 49 elements
QUESTION [5 upvotes]: I am working on this problem from Herstein's book, Abstract Algebra, 3rd edition.
One question asks to construct a field of 49 elements. It gives a hint to use the ring of Gaussian integers and a maximal ideal. This is what I have so far.
All rings are assumed to have a unit and be commutative.
Let R=Z[i] , M:= {a+bi| 7|a and 7|b}
Its pretty easy to show M is an ideal of R.
To show M is maximal, suppose $\exists$N$\subset$R, where N is an ideal and M$\subsetneq$N. So $\exists$a+bi$\in$N s.t 7 $\not|$a or 7$\not|$b. 
so using modular arithmetic we get that $$a^2 + b^2\equiv 1,2,3,4,5,6\pmod7$$.
Therefore letting t = $a^2 + b^2 $ , $\exists$ p,q $\in$$\mathbb{Z}$ such that  7p + tq = 1 
Since N is an ideal and 7 and t are in N, 1 is also in N so it is the whole ring.
So M is maximal and R/M is a field. To show that it has 49 elements, I dont know if what I'm doing is correct. I constructed a group homomorphism from the additive group property of R/M to the group $\mathbb{Z}$$_7$$\times$$\mathbb{Z}$$_7$
$\phi$:R$\rightarrow$$\mathbb{Z}$$_7$$\times$$\mathbb{Z}$$_7$
a+bi $\rightarrow$ ($\bar a$, $\bar b$).
This map is surjective and a group homomorphism
The kernel of this map is also M. So by the first homomorphism theorem for groups I get that R/M $\cong$ $\mathbb{Z}$$_7$$\times$$\mathbb{Z}$$_7$
as groups, therefore |R/M| = 49
Is it correct to do this, view the field as a group and construct a function to find out how many elemnts it has?

REPLY [2 votes]: I can't see anything wrong with what you' ve done, although it would seem to me to be more straightforward to count the elements of $\ R/M\ $ simply by observing that
$$
R/M=\left\{a+ib+M\,|\,a,b\in\{0,1,2,3,4,5,6\}\,\right\}\ .
$$
It's probably also worth noting that there's an obvious field isomorphism between $\ R/M\ $ and a more conventional representation, $\ \mathbb{Z}_7[x]\left/\left\langle x^2+1 \right\rangle\right.\ $, of the field of order $49$, given by
$$
a+ib+M\leftrightarrow a+bx + \left\langle x^2+1 \right\rangle
$$