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- We have seen that the polynomials
*x*^{2}+ 1,*x*^{2}+ 2*x*+ 2 ∈**Z**_{3}[*x*] are irreducible. Find a third irreducible monic quadratic polynomial and prove that the field it defines is isomorphic to the field determined by the other two.

The polynomials*x*^{3}+*x*^{2}+ 1,*x*^{3}+*x*+ 1 ∈**Z**_{2}[*x*] are irreducible. Find an isomorphism between the two fields of order 8 they determine. - Prove that if
*p*is prime there are^{1}/_{2}*p*(*p*- 1) irreducible quadratic monic polynomials in**Z**_{p}[*x*].

Find a formula for the number of irreducible cubic monic polynomials. - Let
*F*be a field of characteristic*p*. Prove that the map*r*↦*r*^{p}is a ring homomorphism.

[You will need to use the fact that if*p*is prime the binomial coefficient_{p}*C*_{r}is divisible by*p*for 0 <*r*<*p*. Can you prove this?]

The above is called the*Frobenius map*introduced by the German mathematician Georg Frobenius (1849 to 1917). - Let
*f*be a ring homomorpism from*R*onto*S*with kernel the ideal*I*. Prove that*f*defines a one-one correspondence between the ideals of*R*which contain*I*and the ideals of*S*.

This result is called the*Correspondence Theorem for Rings*. - Let
*R*be the ring of 2 × 2 matrices with entries in a field*F*. Verify that . Find other similar expressions and deduce that the two-sided ideal generated by a single matrix is either {0} or the whole ring.

A ring with only these two ideals is called*simple*. - Find an element with multiplicative order 8 in the field
**Z**_{3}[x]/ < x^{2}+ 1 > and deduce that the ring of units is indeed cyclic.

Find the inverse of the element*x*in the field**Z**_{3}[*x*]/ <*x*^{3}+ 2x + 1 > of order 27.

Hence or otherwise find the multiplicative order of the element*x*in this field.

How many elements with multiplicative order 26 are there in this field? - Let
*R*be the ring**Z**_{6}[*x*]. How many roots does the polynomial*x*^{2}+*x*have in*R*? How can this happen with a quadratic? *Fermat's Little Theorem*states that if p is prime then*a*^{p}=*a*mod*p*for any integer*a*. Prove that the polynomial*x*^{p}-*x*has*p*roots in**Z**_{p}.

Prove that the polynomial*x*^{pk}-*x*has*p*^{k}roots in a field with*p*^{k}elements.

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