All Questions page - 66

Let \(F\) be a field and \(f(x) \in F[x]\)] be a polynomial of degree 2 or 3. Show that \(f(x)\) is irreducible over \(F\) if and only if \(f(x)\) has no root in \(F\). Give an example to show that the same is not true if deg \(f(x) \ge 4.\)

Show that the polynomials \(2x^4 +6x^3-9x^2+15\) and \(x^6+x^3+1\) are irreducible over \(\mathbb{Z}\) .

Let \(M\) be an \(R\)-module and \(x \in M\) be such that \(rx=0\) for any \(r \in R\) implies that \(r=0\) . Then show that \(R_x \cong R\) as \(R\)-modules.

Let \(R\) be an integral domain and let \(x \in R \backslash\{0\}\) . Show that \(R \cong Rx\) as R-submodules. But \(R \cong Rx\) as rings if and only if \(x\) is a unit in \(R\).

Let \(R\) be a ring. Show that an \(R\)-module \(M\) is a simple module if and only if \(M \cong R \backslash I\) for some maximal left ideal \(I\) of \(R\) .

Let \(R\) be a commutative ring with identity, and let \(e \neq 0,1\)  be an idempotent in \(R\). Prove that \(Re\) cannot be a free \(R\)-module.

Prove that the direct product of a finite number of R-free modules is an R-free module.

Show that every ideal of \(\mathbb{Z}\) is a free \(\mathbb{Z}\)-module .

Let \(R\) be a commutative ring. Prove that \(HOM_R(R,M)\) and \(M\) are isomorphic as left \(R\)-modules

Let \(R\) be a ring. Give an example of a map from one \(R\)-module to another which is a group homomorphism but not an \(R\)-module homomorphism.

An element \(m\) of the \(R\)-module \(M\) is called a torsion element if \(rm=0\) for some non-zero element \(r \in R\) .

(i) Prove that if \(R\) is an integral domain then \(T\) or \((M)\) , the set of torsion elements of \(M\) , is a submodule of \(M\) (called the torsion submodule of \(M\)).

(ii) Give an example of a ring \(R\) and an \(R\)-module \(M\) such that \(T\) or\((M)\) is not an \(R\)-submodule.

(iii) If \(R\) has zero-divisors, show that every non-zero \(R\)-module has non-zero torsion elements.

An \(R\)-module \(M\) is called a torsion module if for each \(m \in M\) there is a non-zero element \(r \in R\) such that \(rm =0\), where r may depend on \(m\). Prove that every finite abelian group is a torsion \(\mathbb{Z}\)-module. Give an example of an infinite abelian group that is a torsion \(\mathbb{Z}\)-module.

Let \(R\) be an integral domain. Prove that every finitely generated torsion \(R\)-module has a non-zero annihilator. Give an example of a torsion \(R\)-module whose annihilator is the zero ideal

Let \(K/F\) be an extension of fields. Let \(\alpha ,\beta \in K\) be algebraic over \(F\) and the degrees of the minimal polynomials of \(\alpha\) and \(\beta\) over \(F\) are \(m\) and \(n\), respectively. Show that if \((m,n) =1\), then \([F(\alpha ,\beta ):F] =mn\) .

Let \(K/F\) be an extension of fields. Show that if every ring \(R\) with \(F \subseteq R \subseteq K\) is a field, then \(K\) is an algebraic extension of \(F\) .

Find the degree and a basis for \(\mathbb{Q}( \sqrt{2} , \sqrt{3} )\) over \(\mathbb{Q}\) .Also show that \(\mathbb{Q}( \sqrt{2} , \sqrt{3} ) = \mathbb{Q}( \sqrt{2} + \sqrt{3} )\)

Find the degree and a basis for \(\mathbb{Q} ( \sqrt{3} + \sqrt{5} )\) over \(\mathbb{Q} ( \sqrt{15})\) .

Prove that \([ \mathbb{Q} ( \sqrt {3-4i} \; + \; \sqrt{3 + 4i} ) : \mathbb{Q} ]=1\) .