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    ok here's the 2nd one:
    Deduce which of the following are fields:
    (i) \mathbb{F}_5(x)/(x^2 + 2x + 2)
    (ii) \mathbb{F}_5(x)/(x^2 + 2)
    (iii) \mathbb{F}_5(x)/(x^2 +3)
    (iv) \mathbb{F}_5(x)/(x^2 + 4)

    Show that \mathbb{F}_5(x)/(x^2 + 2) and \mathbb{F}_5(x)/(x^2 + 3) are isomorphic as rings.

    for part (i) to (iv), im guessing i have to draw out their additive and multiplicative tables and deduce that all polynomials have remainder inverses for them to be fields? is this correct? or are there any other suggestions? thnx

    You should be thinking about irreducability here.

    I feel I have to make a general comment here: you are asking a lot of questions on this topic, and to be honest, the answers to many of them should be obvious from your lectures or textbooks. At this point, my feeling is you are looking for help far too quickly: you should spend more time trying to solve these yourself before asking here.

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