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Since i am not familiar with typing up mathematics using tex or anything so that i can post on the forums, i will use the following notations for ease of readings.

Let F^

*M*denotes

*F*with a superscript M

Let

*F*p^

*M*denotes

*F*^

*M*with suscript p, where p is an odd prime.

There is one part to my problem sets which i am having difficultty constructing examples. There are 3 parts to the question but i can't figure out the last part. Here it is:

Let

*F*be a field. Suppose that

*M*is a nonzero element of

*F*. Let F^

*M*={(a,b) such that a, b both belongs to

*F*}. Define

(a1, b1)*(a2, b2)=(a1b2+b1b2

*M*, a2b1+a1b2)

(a1, b1)+(a2, b2)=(a1+a2, b1+b2), a1, b1, a2, b2 are all elements of

*F*.

a) Suppose that (

*a*^2)-

*M*is not zero for all

*a*belonging to

*F*. Then

*F*^

*M*is a field. Prove the following field axioms hold for

*F*^

*M*

associativity of multiplication

existence of multiplicative identity

existence of multiplicative inverse for nonzero elements

b) suppose that

*a*^2=

*M*for some

*a*in

*F*. Prove that

*F*^

*M*is not a field by demonstrating how one axiom in the definition of field fails to hold.

c) Let

*p*be an odd prime. Prove that there exists a finite field that contains

*p*^2 elements. (Hint: first, show that there exists

*M*in

*F*p such that (

*a*^2)-

*M*is nonzero for all

*a*in

*F*p. according to part a),

*F*p^M is a field. Show that

*F*p^M contains

*p*^2 elements.

Part a) i solved, for multiplicative identity, (a1, b1) has to multiplied by (1,0) to work

For the mulitplicative inverse,

*M*=1

For part b) if

*M*does not equal to 1, then the axiom for the existence of multiplicative inverse fails

for part c) i do not know how to construct practical examples to show me what is actually going on. to show that (

*a*^2)-

*M*is nonzero in

*F*p, do i have to take into account of how the addition and multiplication operations in

*F*^

*M*are defined. And in

*F*p, how can it have

*p*^2 elements? For the last part of part c) how do i take into account that

*F*p^

*M*is in (mod p) with the predefined arithimetic operations above. I mean how do i carry modular arithimetic with such messay mulitplications, and then how can the p^2 elements be listed?

If any of this is not clear, the link to the problem set is here:

http://www.math.utoronto.ca/murnaghan/courses/mat240/ps1.pdf

it is question 9 (c)

I changed alpha to

*M*here.

I am not asking for a solution, but rather how to construct examples so that i can see what is going on in order to solve the question. Thanks everyone for any assistance/suggestions you can give me.