Is the set of even integers a field?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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A field is formed by a set of numbers which have satisfy certain conditions.

These include:

  • Closure with respect to addition to addition and multiplication: This is true for the set of even integers as for any two elements a and b, the element c = a+b and c = a*b lie in the set.
  • Associativity and commutativity for addition and multiplication holding for all elements: This is true for the set as for any three elements a,b and c we have a + (b + c) = (a + b) + c, a*(b*c) = (a*b)*c, a + b = b + a and a*b = b*a
  • Additive and multiplicative inverse lying in the set: The additive inverse of any element a or -a also lies in the set but the multiplicative inverse 1/a for every element a does not lie in the set, e.g. 1/2 does not lie in the set, though 2 does.

Therefore we cannot say that the set of even numbers is a field.

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