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Elements Of Modern Algebra
- 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.arrow_forwardProve that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.arrow_forwardProve Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .arrow_forward
- Let 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 Z7arrow_forwardTrue 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 .arrow_forwardProve that if R is a field, then R has no nontrivial ideals.arrow_forward
- 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.arrow_forwardIf 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 .arrow_forwardLet be a field. Prove that if is a zero of then is a zero ofarrow_forward
- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.arrow_forwardSince this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,