Iff (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such that f (x) has n roots in K and [K: F]Sn !. proof Handwriting solutions
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- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.
- Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field of .
- Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over overLet Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in