Problem 4. Let K be a field and let f ∈ K[x]. Show that if 1+f2 has a factor of odd degree in K[x] then there is an a ∈ K such that a2 = −1.
Rings and fields- Abstract Algebra
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q.
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
let k be a field. 4. Conclude that a factorization of a polynomial f(x) of positive degree as g(x)h(x) is nontrivial iff the factors g(x) and h(x) have degrees strictly less than the degree of f(x).
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting field of p(x), and denote the roots of p(x) in K by α1, α2, α3. a) (10’) If char(F ) does not equal 2, 3, prove that K = F (α1 − α2).
- Let F be a field. Prove: For all (o), 9(a), (x) € F2, if f(x) and (w) are relatively prime and (a)/(x), then (a) and f(a) are relatively prime.
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K] < oo.
1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K]
Let f be the polynomial f()25. Show that f has a parabolic fixed point at the origin, and that f2 has a multiple fixed point at the origin. By calculating fo2, show that f has 4 attracting petals.
Let f be the polynomial f()25. Show that f has a parabolic fixed point at the origin, and that f2 has a multiple fixed point at the origin. By calculating fo2, show that f has 4 attracting petals.
Let F be a field of characteristic p > 0. Show that f = t4 +1 € F[t] is not irreducible. Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
Example 1 provided for reference.
Let K= {0, 1,RX+1} be the four-element field constructed in Example 1 on 206-207. Write X2+X+ 1 as a product of factors of degree 1 in K[X] Example 1 The polynomialx) X2+ X+1 is irreducible in Za[XI, since it has no roots in Z2. Thus (X)) is a maximal ideal in Z,[X), and Z[X]/(f(X is a field. Let us denote it by K. To see what K looks like, notice that the coset g(X) determined...