- #1
Mr Davis 97
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Homework Statement
How many maximal ideals generated by quadratic polynomials does ##\mathbb{Z}_3 [x]## contain?
Homework Equations
The Attempt at a Solution
Here is my reasoning. There are a total of ##2 \cdot 3 \cdot 3 = 18## quadratic polynomials in ##\mathbb{Z}_3 [x]##. If we write ##(x+a)(x+b)##, then there are ##\binom{3}{2} = 3## ways to choose ##a## and ##b##, up to the order of the factors. Since these are the only ways to factor a polynomial, there are 18 - 3 = 15 irreducible quadratic polynomials.
Is this correct?