Suppose that
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Chapter 8 Solutions
Elements Of Modern Algebra
- Prove 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 If and are relatively prime polynomials over the field and if in , then in .arrow_forward8. Prove that the characteristic of a field is either 0 or a prime.arrow_forward
- Let be a field. Prove that if is a zero of then is a zero ofarrow_forwardUse Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .arrow_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_forward
- 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 .arrow_forwardProve that if R is a field, then R has no nontrivial ideals.arrow_forwardLet 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_forward
- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inarrow_forwardProve the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,