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Math Help - Question on Fields

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    Question on Fields

    Show that q is a prime number iff Z/qZ is a field.

    Much appreciated!
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  2. #2
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    Quote Originally Posted by h2osprey View Post
    Show that q is a prime number iff Z/qZ is a field.

    Much appreciated!
    If q is prime then the equation ax\equiv 1(\bmod q) is always solvable for a\not \equiv 0(\bmod q) and so it is a field. Conversely, if \mathbb{Z}/q\mathbb{Z} and suppose q is not prime, then q = q_1q_2 in a non-trivial factorization. But then [q_1]_q[q_2]_q = [0]_q, and that is a contradiction.
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