True or False a. The polynomial f(x) = x^4 - 2x^3 + 6x^2 - 5x + 4 has the factor

Question

True or False a. The polynomial f(x) = x^4 - 2x^3 + 6x^2 - 5x + 4 has the factor x + 2 over the field Q. b. The polynomial f(x) of statement 1a is irreducible over the field R of real numbers. c. Over the field C of complex numbers the polynomial h(x) = x^5 + x^4 - 6x + 2x - 12 has 5 roots counting repeats. d. If K greaterthanorequalto F is a tower of fields with [K: F] = 13 then there is no field E with K > E > F. e. The number pi is algebraic over the field Q. f. For every integer n greaterthanorequalto 1 there exists a polynomial of degree n irreducible over the field Q. g. If E = Q(2^1/3) then [E: Q] = 3.

Explanation / Answer

f(x) = x4 -2x3 +6 x2 -5x +4

(a) at Z3

x = 0,1,2

at x = 0 , we get f(0) = 1

x = 1 , f(1) = 1

x = 2 ,f(2) = 0

at x = 2 f(x) = 0 , f(x) is reducible in linear factors

so x+2 is factor of f(x)

(b) false , because every polynomial of degree greater then equal to 2 have root in real number field.

(c) x5 + x4 -6x +2x -12

has 5 root in complex field in which 4 root are complex and 1 root is real. and 4 complex roots are in paired form with their conjugate

so no counting repetition of roots.

(d) true

(e) false , because pi is transidental number means number which is not root of any polynomial.

(f) true ,because there exist atleast one polynomial of degree n which is irreducible over Q

(g) true

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