Ring theory
Not what you're looking for?
1. If E, F are fields and F is a subring of E, show each q in Aut(E/F) permutes the roots in E of each nonzero p(x) in F[x]. Hint if p(x)=a0+a1x+a2x^2+. . . +anx^n then p(x) has at most n roots in E. show that for z in E, p(z)=0 implies p(q(z))=0
2 If R is a commutative ring of prime characteristic p, show the function f:R-->R, f(r)=r^p is a ring homomorphism
3. 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
4. For the cyclic group (Z,+) = <1> prove the ideal of Z is principal
Purchase this Solution
Solution Summary
This posting exemplifies Ring theory.
Solution Preview
1. Suppose, q is an automorphism of E/F, that is, q(f)=f for all f in F. Let p(x) be a polynomial in F[x], and z be an element of E. Then, p(z) is an element of E, and we can consider q(p(z)). Let p(x)=a0+a1x+a2x^2+. . . +anx^n. Then
q(p(z))=q(a0)+q(a1z)+q(a2z^2)+...+q(anz^n)=a0+a1q(z)+a2[q(z)]^2+...+an[q(z)]^n
Suppose further that p(z)=0. Then, we have
q(p(z))a0+a1q(z)+a2[q(z)]^2+...+an[q(z)]^n=q(0)=0
and so q(z) is again a root ot p(x). Since q is an automirphism, it must be one-to-one, and since p(x) has at most n roots, by the pigeonhole principle, q just permutes the roots of p(x).
2. In a ring of prime characteristic p we have ...
Purchase this Solution
Free BrainMass Quizzes
Probability Quiz
Some questions on probability
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.