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If a is rational and x - a divides an integer monic polynomial, prove that a must be an integer.
Let F be a field of real numbers. Prove that F[x] / (x2 + 1) is a field isomorphic to the field of complex numbers.
Let D be a Euclidean ring, F its field of quotients. Prove the Gauss lemma for polynomials with coefficients in D factored as product of polynomials.
If P is a prime number, prove that the polynomial xn - p is irreducible over the rationals.
A ring of sets is a non-empty class A of sets such that if A and B are in A, then A?B and AnB are also in A. Show that A must also contain A - B.
Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x.
If R is an integral domain, prove that for any two non-zero elements f(x) , g(x) of R[x], deg(f(x)g(x)) = deg f(x) + deg g(x)
Prove that if R is a commutative ring with unit element then R[x] is also a commutative ring with unit element.
If R is an integral domain with unit element, prove that any unit in R[x] is a unit in R and any unit in R is also a unit in R[x].
Let R be a commutative ring with no non-zero nilpotent elements ( that is, an = 0 implies).
If R is a unique factorization domain and if a, b are in R, then a and b have a least common multiple (l.c.m.) in R.
Prove that if R is an integral domain, then R[x1 , x2, .....,xn] is also an integral domain.
Let R={f(x) included in Q[x]: the coefficient of x in f(x) is 0} Prove that R is a subring of Q[x].
If the sequence is split over S, then it is split over R. If the sequence is split over R, then it is split over S.
Show that the class of all finite unions of sets of the form A X B with A ? A and B ? B is a ring of subsets of X ? Y .
Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of O containing the identity.
Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.
Let R = C([0, 1]) be the ring of continuous real-valued functions on the interval [0, 1], with the usual definitions of sum and product of functions.
Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero constant which is a unit in R.
More generally, show that if a divides bc with nonzero a,b then a/(a,b) divides c. (Here (a,b) denotes the g.c.d. of a and b).
A local ring is commutative with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal.
Give a direct proof (i.e. without appealing to part c) that if R is an integral domain, then it is a field.
Show that the inner radius of ring A is squared root 60cm, and find similar expressions for the inner radii of ring B and ring C.
Find elements x and y in a ring of characteristic 4 such that (x + y)^4 != x^4 + y^4.
Indicate which of the following statements about sets under the specified operations are true. For the ones that are false, provide one counter-example.