O. Randal-Williams Lent Term 2017 IB Groups, Rings, and Modules / Example Sheet 3 All rings in this course are commutative and have a multiplicative identity. 1. Show that Z[V-2] and Z[w] are Euclidean domains, where w = >(1+ /-3). Show also that the usual Euclidean function (r) = N(r) does not make Z[V-3] into a Euclidean domain. Could there be some other Euclidean function o making Z[V/-3] into a Euclidean domain? 2. Show that the ideal (2, 1 + /-7) in Z[V-7] is not principal. 3. Give an element of Z[V-17] that is a product of two irreducibles and also a product of three irreducibles. 4. Show that if R is an integral domain then a polynomial in R[X] of degree d can have at most d roots. Give a quadratic polynomial in (Z/8Z)[X] that has more than two roots. 5. Determine whether or not the following rings are fields, PIDs, UFDs, integral domains: Z[X], Z[X]/(x2+1), Z[X]/(2, X2+1), Z[X]/(2, x2 +X+1), Z[X]/(3, X3 - X +1). 6. Determine which of the following polynomials are irreducible in Q[X]: X4 +2X +2, X4 + 18X2 + 24, X3 - 9, X3 + X2 + X +1, X4 + 1, X4 + 4. 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a and b in R is an element d in R such that d divides both a and b, and if c divides both a and b then c divides d. (i) Show that the god of a and b, if it exists, is unique up to multiplication by a unit. (ii) In lectures we have seen that, if R is a UFD, the god of two elements exists. Give an example to show that this is not always the case in an integral domain. (iii) Show that if R is a PID, the god of elements a and b exists and can be written as ra + sb for some r, s E R. Give an example to show that this is not always the case in a UFD. (iv) Explain briefly how, if R is a Euclidean domain, the Euclidean algorithm can be used to find the gcd of any two non-zero elements. Use the algorithm to find the gcd of 11 + 7i and 18 - i in Z[i]. 8. Find all ways of writing the following integers as sums of two squares: 221, 209 × 221, 121 x 221, 5× 221. 9. By working in Z[V/-2], show that the only integer solutions to x2 +2 = y3 are x = +5, y = 3. 10. Exhibit an integral domain R and a (non-zero, non-unit) element of R that is not a product of irreducibles. 11. Let Fq be a finite field of q elements. (i) Show that the prime subfield K (that is, the smallest subfield) of Fg has p elements for some prime number p. Show that Fg is a vector space over K and deduce that q =