Ideals and Rings : Homomorphisms
Problem: Prove the Second Isomorphism Theorem: If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S, respectively, and (S+A)/A isomorphic to A/(S intersecting A).
Problem: Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))
Ideals and Factor Rings : Annihilators and Fermat's Theorem
If R is a commutative ring, a polynomial f(x) in R[x] is said to annihilate R if f(a) = 0 for every a belonging to R Show that x^p - x annihilates Zp (Z is integers)
Systems of Equations : Find Length and Width of a Rectangle
A rectangular yard is surrounded by a fence that is 4 feet longer than it is wide. If the perimeter of the fence is 424 feet, what are the dimensions of the yard it encloses?
I am having diffilculties understanding the whole concept of Real Numbers and Their Properties, Real Numbers, Fractions, Addition and Subtraction of Real Numbers, Exponential Expressions and the Order of Operations, Properties of the Real Numbers and Using the Properties to Simplify Expressions. What I need is a break-down of th ...continues
I am having a little trouble with this proof: If f:A-->B and g:C-->D f=g iff A=C and for every x in A, f(x)=g(x)
Algebraic Identities into Geometrical representations
Draw and describe how each of the following algebraic identities could be represented geometrically: a) (a-b)^2 = a^2 - 2ab + b^2 b) a(b+c)=ab + ac c) (a+b)(c+d)=ac+bc+ad+bd Please use illustrations of each with an explanation of the illustration so I can understand the process used to complete this type of problem. THA ...continues
Pythagorean Theorem Word Problem
"There is a bamboo 10 ft. high, the upper end of which being broken reaches the ground 3 feet from the stem. Find the height of the break." Please show your work in steps so I can understand the process used to solve it.
Upper Level Number Theory Problems
Please see attached document. There are four problems. The first problem is 71! mod 73
Linear independence of embeddings
Let E be a finite extension of a field F. Show that any finite set of distinct embeddings of E into the algebraic closure of F is linearly independent over F.
