University of Utah, Department of Mathematics May 2020, Algebra Qualifying Exam

Show there is no simple group of order \(448=2^6\cdot 7\).

Suppose \(G\) is a finitely generated Abelian group such that the automorphism group of \(G\) is finite. Let \(T\) be the torsion subgroup of \(G\). Describe all possibilities for \(G/T\) up to isomorphism.

Let \(R=\mathbb {Z}[x]\) and suppose that \(I=\langle x\rangle \) and that \(J=\langle x,5\rangle \). How many elements are in \(\operatorname {Tor}^R_i(R/I,R/J)\) for \(i=0,1,2\)?

Suppose \(R=\mathbb {F}_3[x]\). Let \(M\) be the module generated by three elements \(a,b,c\) subject to the three relations \(-xa+x^2b+(x^2-1)c=0\), \(xb+xc=0\) and \(xb+x^2c=0\). Find a direct sum of cyclic \(R\)-modules isomorphic to \(M\).

Show that there are at most 5 groups of order 20.

Let \(K\) be an algebraically closed field. Let \(f\in K[x,y]\) be irreducible and let \(g\in K[x,y]\) be such that \(f\) does not divide \(g\). Prove that there is a point \((k_1,k_2)\in K^2\) such that \[ f(k_1,k_2)=0\text { and }g(k_1,k_2)\ne 0. \]

Find a polynomial \(f(x)\in \mathbb {F}_2[x]\) such that \(\mathbb {F}_2[x]/f(x)\) is a field with 8 elements.

Let \(R\) be the ring of continuous real-valued functions on the closed interval \([0,1]\). What are the maximal ideals of \(R\)?

Let \(\sigma \) be the automorphism of \(\mathbb {F}_5(t)\) sending \(t\) to \(t+1\). Let \(G\) be the subgroup of \(\operatorname {Aut}(\mathbb {F}_5(t))\) generated by \(\sigma \). Find an element \(s\in \mathbb {F}_5(t)\) such that \[ \mathbb {F}_5(t)^G=\mathbb {F}_5(s). \]

Let \(2^{1/4}\) be the unique positive real fourth root of of 2 in \(\mathbb {R}\). Show that \(\mathbb {Q}(2^{1/4})/\mathbb {Q}\) is not a Galois extension.