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How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP ring is $30 and on an SST ring is $60?
Prove that a finite subring R of a field F is itself a field. Hint: if x is an element of R and x is not equal to 0 show the function f:R->R with f(r).
If R and S are rings, the cartesian product RxS is a ring too with operations and additive inverse -(r,s) = (-r,-s)
Suppose F, E are fields and F is a subring of E. Suppose g is an element of E and is algebraic over F, that is p(g)=0 for some nonzero.
If E is a field and H is a subgroup of Aut(E), show that the set E^H of elements of E that are fixed (sent to themselves) by every Q in H.
If a,b are positive integers with greatest common divisor d and least common multiple m, show aZ+bZ=dZ and aZ intersect bZ=mZ.
Let R be a commutative ring and let I ? R be an ideal. Show that vI := { f ? R | there exists n ? N such that fn ? I } is an ideal of R.
We have seen that R is a principle ideal domain. That is, every ideal is generated by a single element of R. Find h(x) in R so that I = {f(x)h(x) | f(x) in R}.
Let p be any prime integer. Consider polynomials f(x) and g(x) of the form. Consider the multiplicative group of nonzero elements of Zp.
How to evaluate personality theory over time, and how and why historical personality theories lend themselves to understanding people and events.
If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding.
In each of the cases below, describe the ring obtained from F2 by adjoining an element x satisfying the relation: (i) x2+x+1=0, (ii) x2+1=0, (iii) x2+x=0.
For the previous rings that you decided are not local, slightly change their definition (without proof) such that you get a local ring.
Identify Hom(Z/nZ,Z), Hom(Z,Z/nZ), Hom(Z/3Z,Z/6Z), Hom(Z/10Z,Z/6Z) as abelian groups, where n belongs to Z and Z is the set of integers.
If f(x) and g(x) are two non-zero elements of F[x], then prove that deg [f(x)g(x)] = deg f(x) + deg g(x)
Prove that the polynomial x2 + x + 1 is irreducible over F, the field of integers modulo 2.
If zero cannot be used in the first position and repeats are allowed in the other 6 positions, how many license plates are possible?
How many ways can 5 Manchester United and 8 Chlesea players be seated at a circular dinner table if no two Manchester United players can sit together?
An airline has 15 flights from city A to city B and eight flights from city B to city C. In how many ways can you fly from city A to city B.
There are two small cities, Santa Clara and Santa Rosa. Santa Clara has 20 houses and Santa Rosa has 15 houses.
A set D is a subset of set C provided that?Provide complete and step by step solution for the question.
Eight people are attending a seminar in a room with eight chairs. In the middle of the seminar, there is a break and everyone leaves the room.
Find a complete set of mutually incongruent solutions. Provide complete and step by step solution for the question.
How many possible social security numbers are there if numbers can be repeated but 0 cannot be used for the first digit?
Preparing a plate of cookies for 8 children, 3 types cookies {chocolate chip, peanut butter, oatmeal}, unlimited amount of cookies in supply.