1. Let G=GL(2,Q), Q meaning rational numbers, and let A =matrix 0 -1 1 0 and B =matrix 0 1 -1 1 Show that A^4 = I = B^6, but that (AB)^n does not equal I for all n >0. Conclude that AB can have infinite order even though both factors A and B ha ...continues
prove: if H is a subgroup of G, and K is a normal subgroup of G, prove that HK=KH
if H is a subgroup of G, and K is a normal subgroup of G, prove that H intersection K is a normal subgroup of H
Prove that element a of a ring R with unity has a multiplicative inverse in R, then a is not a zero divisor in R
Let f:A->B where A and B are nonempty. Prove that f has the property f^-1(f(S))=S for every subset S of A if and only if f is one-to-one
1. a. Is R= {a+b(squareroot of 2): a,b element of Z} a domain? b. Using the fact that alpha= (1/2)(1 + (square root of -19)) is a root of ((x^2)- x + 5), prove that R={a + b(alpha) : a,b element of Z} is a domain. Z= integers 2. Assume that (x-a) divides f(x) in R[x]. Prove that (x-a)^2 divides f(x) if and only if ...continues
1. a. show that every subfield of complex numbers contains rational numbers b. show that the prime field of real numbers is rational numbers c. show that the prime field of complex numbers is rational numbers 2. a. Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero cons ...continues
1. Define F sub four to be the set of all 2x2 matrices. F(sub 4)= [ a b ] ; a,b elements of F sub 2 b a+b i) Prove that F sub four is a commutative ring whose operations are matrix addition and matrix multiplication ii) prove that F sub four is a field having exactly four elements iii) show that I sub f ...continues
Let K be a field of characteristic p > 0, and let c be an element of K. Show that if α is a root of f(x) = x^p − x − c, so is α + 1. Prove that K(α) is Galois over K with group either trivial or cyclic of order p.
Find all subfields of Q(sqrt(2), sqrt(3))with proof that you have them all. What is the minimal polynomial of sqrt(2) + sqrt(3)? Which of your subfields does it generate over Q?
