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- Where Q is the field of rational numbers and i is the imaginary number. Such that i²= -1 Also, Q(√2) = a + √2b Where a and b are rational numbers.(please solve it fast)4) Let F be a field. The derivation f'(T) e F[T] of a polynomial f(T) = Ea; · Ti e F[T] is defined by i20 f'(T) := E(i · a;) - Ti-1. i21 Verify for all f(T) and g(T) in F[T] the rules (F(T)+ g(T))' = f'(T) + g'(T) and (F(T) · g(T))' = f'(T) · g(T) + f(T) · gʻ(T).
- Please help me fast so that I will give Upvote.please do within 20 minutes.Form the interpolating polynomial of degree k that satisfies pi) (xo) = = fG) (xo) for 03) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T) in R[T], which are both not constant, i.e. of degree > 1, such that f(T) = g(T) - h(T).+ aaz", as + 0 be a polynomial with real coefficients 3. Let d 2 2 and let P(z) = ao + aịz + such that Ja;| < 1 for j = 0,1,...,d. The purpose of this exercise is to show the existence of a polynomial f(z) = co + c1z + ...+ caz“, where c; = ±1 for j = 0,1,...,d and f well approximates ... P, i.e., max |P(z) – f(z)| = 0( Vd log d), by employing the probabilistic method based on the next steps. (a) Show that P satisfies |P(z) – P(w)| < d²]z – w[ for all z, w e [-1, 1]. Hint; you may show that |P(z) < d² for all |z| < 1. (b) Show that there exist independent random signs fo. E1....Ea such that EF;] = a; for j = 0, 1, ..., d.Let f e E(0, ox) and let a be a positive real number. Define the function fa as follows Sal) = {"; f(t – a), if t > a, if 0(1) Consider the function f: R\{-1} → R given by f(x) : your claim. I+: I (2) Consider the function f: NxN→Q given by f(x, y) = = 31 + your claim. Find the image of f and prove Find the image of f and prove (3) Let F be a field. Without looking up the proofs from the book, or notes from class/TUT prove that the additive and multiplicative inverses in a field are unique. (4) Prove "cancellation" for fields (without looking up the proofs): (a) For all z, y, z € F, if z + z = y + 2, then x = y. (b) For all z, y, z € F \ {0}, if z-z = y. z, then z = y. (5) Let F = {0, 1, a, b} be a field with 4 elements. Prove that if 1 + 1 = 0, then a + b = 1. (6) Let F be a field and a € F. Prove that -(-2) = 1. (7) Let F be a field. Prove that (-1). z = -x. (8) Let P = "(VER) (y € R) (r+y> 0)". Let Q = "(VER) (Vy E R) [(ry > T) V (zySEE MORE QUESTIONSRecommended textbooks for youElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
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