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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
The product of two complex numbers
Write the equation of the line that passes through point (0, -5) with a slope of 7.
Let epsilon, gamma >0. If x is epsilon-close to y, and y is gamma-close to z, then x and z are (epsilon + gamma)-close. Let epsilon>0. If y and z are both epsilon close to x, and w is between y and z (i.e.. y
Show manual work (by hand) step by step to reach the optimal answer (i.e., don't just provide a final answer and don't use a computer program to find a solution). Provide also a summary of your solution.
Let u be an algebraic element of K (field extension of F) whose minimal polynomial in F[x] has prime degree. If E is a field such that F is contained in E is contained in F(u)
Quadratic expressions
Graph the inequality 2x -3y > 6. Which portion of the grid satisfies the inequality? A line with a slope of -1 has an x intercept of 3. What is the y intercept?
brody rode his bike 70 miles in 4 hours. he rode at an average speed of 17 mph for t hours and at an average rate of
Algebraic expression by addition rule.
nate left for a basketbaal trip and realized that he forgot his shoes. he calledhis mom and she agreed to get his shoes
problemtwo companies start up at the similar time. company a claims their annual profits follow a linear model px10x-7
with regression we are concerned about variability around the rather than variability around the which would be the
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