I am auditing a Number Theory course. One of the question involves the "9 elements of $Z_3[x]_{x^2+1}$". I'm not familiar with this notation (and I can't seem to find it anywhere). What are the 9 elements of this, for lack of a better word, thing (ring?), and how are they found?
2026-04-25 13:08:28.1777122508
What is this notation ($Z_3[x]_{x^2+1}$)? What are the elements of this?
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In standard notation, it should be $(\Bbb Z/3\Bbb Z)[x]/(x^2 + 1)$, i.e. the polynomial ring over the field with $3$ elements, quotiented by the ideal generated by the polynomial $x^2 + 1$.
It is thus the unique (up to isomorphism) quadratic extension of $\Bbb F_3$, hence has $9$ elements.
These elements can be represented by $\{a + bx: a, b \in\{0, 1, -1\}\}$.
Wiki has more details.