Multiple Table for Four Elements

Please provide a detailed solution to the attached problem.

I already found the multiplication table for a field with 4 elements (and showed that the nonzero elements form a cyclic group). Hence for the second part of the question, please only write down a multiplication table for 9 elements (and show that the nonzero elements form a cyclic group).

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..., 2x + 2}

The multiplication table is

1 2 x x + 1 x + 2 2x 2x + 1 2x + 2

1 1 2 x x + 1 x + 2 2x 2x + 1 2x + 2

2 2 1 2x 2x + 2 2x + 1 x x + 2 x + 1

x x 2x 2 ...