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- Show that the set of all subsets of a set
*S*forms a ring under the operations*A*+*B*=*A*∪*B*-*A*∩*B*and*A*.*B*=*A*∩*B*for any subsets*A*,*B*∈*S*. This is called a*Boolean ring*.

[Use Venn diagrams to check the ring-theory axioms.]

Does this ring have an identity element? Which elements of the ring have multiplicative inverses?

If instead we define addition by*A*+*B*=*A*∪*B*do we then get a ring ? - By considering the product of "pure quaternions" (of the form
*bi*+*cj*+*dk*) show how the scalar product and the vector product of vectors in**R**^{3}can be obtained from quaternionic multiplication.Show that the set

**R**^{3}under vector addition and the vector product × is*not*a ring. - Prove that the set of real polynomials
*a*_{0}+*a*_{1}*x*+*a*_{2}*x*^{2}+ ... +*a*_{n}*x*^{n}for which*a*_{0}=*a*_{1}= 0 is a subring of**R**[x]. Is it an ideal?Prove that the set of all real polynomials

*a*_{0}+*a*_{1}*x*+*a*_{2}*x*^{2}+ ... +*a*_{n}*x*^{n}for which the sum*a*_{0}+*a*_{1}+*a*_{2}+ ... +*a*_{n}= 0 is an ideal of**R**[x].

[Hint: such a polynomial satisfies*p*(1) = 0.] - Prove that the set {
*r*+*s*√2 |*r*,*s*∈**Q**} is a field under real addition and multiplication. Prove that it is the smallest subfield of**R**which contains √2. - Let
*R*be the ring of elements of the form {*a*+*bx*|*a*,*b*∈**Z**_{2}} with arithmetic modulo 2 and multiplication using the "rule"*x*^{2}= 1. Prove that this is*not*a field.Let

*R*be the ring of elements of the form {*a*+*bx*|*a*,*b*∈**Z**_{3}} with arithmetic modulo 3 and multiplication using the "rule"*x*^{2}=*x*+ 1. Prove that this is a field with 9 elements. - Show that in any finite field the additive order of any non-zero element must be a prime.

Prove that in any finite field the additive orders of any two non-zero elements are the same. (This is called the*characteristic*of the field.) - Let
*p*be a prime number and*k*∈**N**. Consider the set*R*= {*a*_{0}+*a*_{1}*x*+ ... +*a*_{k-1}*x*^{k-1}|*a*_{1},*a*_{2}, ... ,*a*_{k}∈**Z**_{p}} with arithmetic modulo*p*and multiplication using the "rule"*x*^{k}=*q*(*x*) for some fixed polynomial*q*(*x*) ∈**Z**_{p}[*x*]) of degree <*k*.

Prove that*R*is a ring with*p*^{k}elements.

If the polynomial*p*(*x*) =*x*^{k}-*q*(*x*) can be factorised in**Z**_{p}[*x*] as a product of two lower degree polynomials*r*(*x*) and*s*(*x*), prove that*R*has zero-divisors.Prove that the polynomial

*p*(*x*) =*x*^{2}-*x*- 1 of Question 5 cannot be written as a product of linear factors in**Z**_{3}[*x*].

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