Recent content by kathrynag

Right idea?

1. Let K be a finite field of characteristic p > 0. Show that the map f : K--->K given by f(a) = a^p is surjective, hence if B is in K, then N = l^p for some element l of K. 2. Let K be a finite field with q = p^n elements (p a prime). Show that if f(x) in K[x] and l is a root of f(x) in...

Consider g(x)=x^2sin(1/x) if x>0 and 0 if x<=0 1. a) Find g'(0) b) Compute g'(x) for x not 0 c)Explain why, for every delta>0, g'(x) attains every value between 1 and -1 as ranges over the set (-delta,delta). Conclude that g'(x) is not continuous at x=0. Next, we want to explore g with...

2. I was thinking of somehow using a theorem stating [F:K]=[F:K(u)][K(u):K] based on the deg m(x) being odd, I would say deg m(x)=2n+1

Is this a good way to start or should I try something different?

1.Find the degree and basis for Q(3^1/2,7^1/2) over Q. 2.For any positive integers a, b, show that Q(a^1/2+b^1/2)=Q(a^1/2,b^1/2) Ideas: 1. Well I know if I looked at (3)^1/2 over Q Then (3)^1/2 has minimal polynomial x^2-3, so degree 2 over Q (7)^1/2 has minimal polynomial x^2-7 so...

1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2. 2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2). 1. I was thinking of...

3. We have lim_{c^{-}}\neqlim_{c^{+}}. Would I maybe use upper and lower sums to show not integrable?

Ok, think I have a good start on 1 and 2 now. Are there any hints on starting 3 and 4?

For 2. I looked at integral(f+tg)^2 integral(f^2+2tgf+t^2g^2).

Thanks a lot! I was posting from my Ipod so I couldn't get the Tex.

Integration 1.Let f be integrable on [a; b] for every b > a; where a is fixed. Define integralf(x)dx(bonds on integral(a,infinity) = limb-->infinity(integralf(x)dx)(bounds a,b); provided the limit exists. Prove the so called integral test: if f(x) >= 0 and if f decreases monotonically...

2. elements of F have minimal polynomials of K If E/K is a subfield of F/K, then we have a minimal polynomial of E and thus a minimal polynomial of E of K. So algebraic? 3.Do I look at 1=u*m(x)? 4. We have degree=m<infinity I want ot do something like this: [M:K]=[M:L]*[L:K]. but we...

For 2, since F(u)=0, and F is contained in E, could i say F(u) is contained in E?

1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E. Let a be any element of E that is not in K. Then a = f(u)/g(u) for some polynomials f(x), g(x) inK[x] 2.Let K contained in E...