Problem:
Q1. 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
Q2 If R is a commutative ring of prime characteristic p, show the function f:R-->R, f(r)=r^p is a ring homomorphism
Q3. 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
Q4. For the cyclic group (Z,+) = <1> prove the ideal of Z is principal consist of all rational numbers for which the denominator of the reduced form is odd. Show that this set forms a ring under the usual operations of addition and multiplication.
Provide complete and step by step solution for the question and show calculations and use formulas.