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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?

ldiaz
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    I would personally feel more comfortable finding a degree-2 extension of the field of seven elements. For instance by adjoining a square root of $-1$. – Arthur Jan 09 '20 at 07:14
  • Usually I would do something with some irreducible polynomial but this book doesn't even introduce polynomial rings until after this problem. So I'm trying to solve it with whatever has been presented in the book so far, which isn't much if I'm being honest – ldiaz Jan 09 '20 at 07:18
  • why $7$ does not divide both $a$ and $b$? – MANI Jan 09 '20 at 07:33
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    @MANI ah, my mistake, it’s supposed to be an OR statement, I just fixed it – ldiaz Jan 09 '20 at 07:34

1 Answers1

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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 $$ a+ib+M\leftrightarrow a+bx + \left\langle x^2+1 \right\rangle $$

lonza leggiera
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