Give an example, with justification, to show that: 1.the union of integral domains need not be an integral domain; 2.all finite fields are not isomorphic.?

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Feb 10, 2018

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Example of two integral domains whose union is not one

A rich source of examples of such things is matrices.

For example, note that:

#((0, 1), (2, 0))^2 = ((0, 1), (2, 0)) ((0, 1), (2, 0)) = ((2, 0), (0, 2))#

That is #((0, 1), (2, 0))# is a square root of #2#.

We can identify an integral domain of matrices of the form:

#m((1, 0), (0, 1)) + n((0, 2), (1, 0)) = ((m, n), (2n, m))#

with #m, n in ZZ# which is isomorphic to the ring of numbers of the form #m+n sqrt(2)#.

Note that this has no zero divisors since #sqrt(2)# is irrational.

Similarly, we find:

#((0, 2), (1, 0))^2 = ((2, 0), (0, 2))#

So #((0, 2), (1, 0))# is also a square root of #2# and we can identify an integral domain of matrices of the form

#((m, 2n), (n, m))#

which is also isomorphic to the ring of numbers of the form #m+n sqrt(2)#

The union of these two sets of matrices is not closed under addition and therefore not a ring, let alone an integral domain.

For example:

#((0, 1), (2, 0)) + ((0, 2), (1, 0)) = ((0, 3), (3, 0))#

is not in either of the two integral domains.

Furthermore, if we add all elements required to make the set of matrices closed under addition and additive inverse, then we get a ring with zero divisors. For example:

#(((0, 3), (3, 0)) - ((3, 0), (0, 3)))(((0, 3), (3, 0)) + ((3, 0), (0, 3)))= ((-3, 3), (3, -3))((3, 3), (3, 3))= ((0, 0), (0, 0))#

Finite fields of different orders or characteristics cannot be isomorphic

For example #GF(2)# and #GF(3)# have #2# and #3# elements respectively, so there is not even a bijection between them as sets, let alone an isomorphism of fields.

All finite fields have #p^n# elements for some prime #p# and positive integer #n#. All finite fields of the same order are isomorphic. That is, a finite field is essentially determined by the values of #p# and #n#.

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