Hello, everyone, i am a newbie here. I am currently taking a modern linear algebra course that also focus on vector spaces over the fields of Zp and complex numbers.(adsbygoogle = window.adsbygoogle || []).push({});

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^MdenotesFwith a superscript M

LetFp^MdenotesF^Mwith 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:

LetFbe a field. Suppose thatMis a nonzero element ofF. Let F^M={(a,b) such that a, b both belongs toF}. Define

(a1, b1)*(a2, b2)=(a1b2+b1b2M, a2b1+a1b2)

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

a) Suppose that (a^2)-Mis not zero for allabelonging toF. ThenF^Mis a field. Prove the following field axioms hold forF^M

associativity of multiplication

existence of multiplicative identity

existence of multiplicative inverse for nonzero elements

b) suppose thata^2=Mfor someainF. Prove thatF^Mis not a field by demonstrating how one axiom in the definition of field fails to hold.

c) Letpbe an odd prime. Prove that there exists a finite field that containsp^2 elements. (Hint: first, show that there existsMinFp such that (a^2)-Mis nonzero for allainFp. according to part a),Fp^M is a field. Show thatFp^M containsp^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) ifMdoes 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)-Mis nonzero inFp, do i have to take into account of how the addition and multiplication operations inF^Mare defined. And in

Fp, how can it havep^2 elements? For the last part of part c) how do i take into account thatFp^Mis 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 toMhere.

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.

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# Questions concerning Finite Fields (Basic Abstract Algebra)

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