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Elements Of Modern Algebra
- Show that the ideal is a maximal ideal of .arrow_forwardSuppose 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 Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .arrow_forward
- 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_forwardFind the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.arrow_forward29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .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_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardFactors each of the polynomial in Exercise 1316 as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers. 6x4+x3+3x214x8arrow_forward
- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.arrow_forwardLet be a field. Prove that if is a zero of then is a zero ofarrow_forward1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. andarrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,