- #1
kathrynag
- 595
- 0
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 some extension F of K, then l^q is also a root of f(x).
3. Let f(x) in Zp[x] be irreducible of degree m. Show that if f(x) | x^p^n− x, then m | n.
4. Let K be a field with p^m elements and let l in K. Show that l^p^mk= l for every
positive integer k.
Ideas:
1. Since K is a finite field, f maps K to itself, f is surjective iff f is injective.
So I need to show f is injective, I'm assuming?
2.I began by writing f(x) = a_mx^m+a_m−1x^m−1+· · ·+a_1x+a_0. I was thinking that might help somehow.
3.Let f(x) | x^p^n− x and let K be the field of p^n elements. Then x^p^n− x,
and hence f(x), splits over K. I get this, but then I don't know where to go from here.
4.I think we can use induction somehow?
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 some extension F of K, then l^q is also a root of f(x).
3. Let f(x) in Zp[x] be irreducible of degree m. Show that if f(x) | x^p^n− x, then m | n.
4. Let K be a field with p^m elements and let l in K. Show that l^p^mk= l for every
positive integer k.
Ideas:
1. Since K is a finite field, f maps K to itself, f is surjective iff f is injective.
So I need to show f is injective, I'm assuming?
2.I began by writing f(x) = a_mx^m+a_m−1x^m−1+· · ·+a_1x+a_0. I was thinking that might help somehow.
3.Let f(x) | x^p^n− x and let K be the field of p^n elements. Then x^p^n− x,
and hence f(x), splits over K. I get this, but then I don't know where to go from here.
4.I think we can use induction somehow?