Let [g] be an element of order 180 in Z181- (a) Prove that every non-zero element [a] of Z181 can be written as a power of g.

Q: Let G=Z5xZ6X Z9 X Z12- Find the elementary divisors and invariant factors of G.

A: Solution :-

Q: Let p be a prime. Are there any nonconstant polynomials in Zp[x]that have multiplicative inverses?…

A: Given: A prime number p. To check: Whether any nonconstant polynomials in Zp[x] have multiplicative…

Q: Find all possible pairs (a, b) in Z3 × Z3 such that Z3[r]/(r³+ar+b) is a field.

A:

Q: Let [g] be an element of order 180 in Z181. Justifying your answer, say whether [19] is a primitive…

A: Here given ℤ181 is the addition modulo group of order 181. It is a cyclic group and 181 is a prime…

Q: Find a monic polynomial of least degree in R[x] that has 1-i and 2i as roots. write the powers of x…

A:

Q: Prove that G1 ⨁ G2 is isomorphic to G2 ⨁ G1. State the general case

A: To Prove: G1⊕G2 is isomorphic to G2⊕G1. Also, state the general case.

Q: Let C([0, 1]) denote the set of all continuous functions f : [0, 1] → R, and let Qn (0, 1] =…

A:

Q: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Findthe multiplicative inverse of α +…

A: Let α be a zero of x3+x2+1 in some extension field of ℤ2. We need to find multiplicative inverse of…

Q: De ne function f : Z ! N by f(n) = 2n if n > 0 and f(n) = 1 ? 2n if n   0. (a) Observe that the…

A:

Q: Prove that no element of ℤ/n is both a zero divisor and a unit.

A: To Determine :-  Prove that no element of ℤn is both a zero divisor and a unit.

Q: To submit Let R[] be the set of all expressions a = ao+ajx+a2x²+ · where aj e R for all nonnegative…

A:

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

A:

Q: The fixed field of G is a subfield of 7.

A: Statement : The fixed field of G is a subfield of F.We have to prove this statement.

Q: Find the minimal polynomial for  ∛2 + ∛4  over Q.

A:

Q: Verify that the commutator of two derivations of an F-algebra is again a derivation, whereas the…

A:

Q: Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R[x] by defining an+1,&n+2,...…

A:

Q: Let a be a non-zero element of Z43. w| 42. Show that [3] is a primitive root of Z43 and that [2] is…

A:

Q: Use a purely group theoretic argument to show that if F is a fieldof order pn, then every element of…

A:

Q: Find all values of p such that Z„[x]/(x² + 1) is a field.

A: Given problem is :

Q: LET N[x] be the set of all polynomials with coefficient in N. Prove that N[x] is countably infinite

A: Finite product of countably infinite set is countably infinite  And Countable union of countably…

Q: Let N[x] be the set of all polynomials with coefficients in N. Prove that N[x] is countably…

A: Let N[x] be the set of all polynomials with coefficients in N. Then we have to show that N[x] is…

Q: Use a purely group theoretic argument to show that if F is a fieldof order pn, then every element of…

A: As per the question we have to use pure group theoretic arguments to show that if F is a field of…

Q: a) Factor a + r³ + x into irreducible polynomials in Z2[x];

A: We shall solve first Que only, if you want others to be solved kindly post it again by mentioning…

Q: Let 5 be a primitive 9th root of unity in C and consider the extension Q(5) of Q. Prove that T; is a…

A: The Galois extension field is very significant in field theory. For a primitive nth root of unity,…

Q: Let f=z" + an-12¹-1 + ... + a₁ € C[z] be a polynomial of degree n ≥ 1. Which of the factorisations…

A: Solution:-

Q: et [g] be an element of order 180 in Z181. Let [a] be a non-zero element of Z181, what are the…

A:

Q: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Find the multiplicative inverse of α +…

A: As per the question we are given a polynomial p(x) = x3 + x2 + 1 and α is a root of p(x) = 0 in some…

Q: Find all monic irreducible polynomials of degree 2 over Z3.

A:

Q: Show that R[x]/<x2 +1> is a field.

A: To show that ℝx/x2 + 1 is a field, we enough to show that x2+1 is maximal in ℝx. Suppose that I =…