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).
Let k be a field. 4. Conclude that a factorization of a polynomial f(x) of positive degree as g(x...
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...
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).
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()...
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...
Question 4 (20 points) Let F: R R be any homogeneous polynomial function (with degree no less than one) with at least one positive value. Prove that the function f:Rn R, f(x) F(x) 1, defines on f-1(0) a structure of smooth manifold.
Let f(x) = do +212 + ... + and" be a polynomial of degree less than or equal to n, and let {x0, 21, ..., Un} be distinct points. What is the value of f (x0, X1, ... , Xn]? Explain.
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of all polynomials of degree less than or subring of Ra (iii) Find the quotient q(x) and remainder r(x) of the polynomial P\(x) 2x in Z11] equal to k. Is Rr]k a T52r43 -5 when divided by P2(x) = iv) List all the polynomials of degree 3 in Z2[r]. subring of the polynomial ring R{z]...
let fx be a polynomial of degree <= to n whats the value of f(Xo, X1....Xn). explain Let f(x) = ao tai xt...... + Anxh be a polynomial of | degree less than or equal to n, and let {xo.xi... n} be distinct points What is the value of f[xo, X.. Xn] Justify / Explain.
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
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.