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​This question asks about using the Sage functionality for computing in Finite Fields.​

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  1. Use Sage to create a finite field with 17 elements
    In this field calculate:
    The difference: 13 – 16
    The sum: 11 + 10
    The quotient: 1/2
    The product: 3 * 8
    The multiplicative inverse of: 5
  2. Use Sage to create a finite field with 32 elements. Let ‘a’ denote the primitive element.
    In this field Calculate:
    The difference: (a^2 + a) – (a + 1)
    The multiplicative inverse of: a^4 + a + 1
    The quotient (a^2 + 1)/(a^4 + a + 1)
  3. Use Sage to create a finite field with 5^3 elements. Let ‘alpha’ denote the primitive element.
    In this field Calculate:
    The sum: (3*alpha^2 + 4*alpha) – (alpha^2 + 3)
    The multiplicative inverse of: (alpha + 1)
    The product: (alpha + 2)*(alpha + 3)
  4. Use sage to create a finite field with 503,777,509 elements.
    In this field calculate:
    The quotient: 123,456,789/456,555,333
    The multiplicative inverse of : 987,654,321
    The difference: 789,123,456 – 444,333,111

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