University of Utah, Department of Mathematics January 2018, Algebra Qualifying Exam

Show that there is no simple group of order 112. You may use the fact that \(A_7\) is simple.

Describe the automorphisms of the group \((\mathbb {Z}/2\mathbb {Z})\times (\mathbb {Z}/4\mathbb {Z})\).

Prove that there is no element in \(S_9\) of order 18.

Let \(M\) be the \(\mathbb {C}[x]\)-module generated by elements \(a,b,c\) modulo the three relations \(a+b\), \(x^3a+xb+xc\) and \((x^3+1)a+b\). Write \(M\) as a direct sum of cyclic \(\mathbb {C}[x]\)-modules.

Let \(R=\mathbb {Q}[x,y,z]\) and let \(I=(x,y)\) and \(J=(y,z)\). Compute \(\operatorname {Tor}^R_i(R/I,R/J)\) for all values of \(i\ge 0\).

Find all prime ideals of the ring \(\mathbb {Z}[x]/(15,x^3-2)\).

Prove that if a \(3\times 3\) matrix \(A\) over \(\mathbb {Q}\) satisfies \(A^8=I\), then \(A^4=I\). Justify claims of irreducibility of polynomials.

Let \(F\subset L\) a extension of fields of degree 4. Prove that there are no more than 3 fields proper intermediate subfields \(K\); namely, such that \(F\subset K\subset L\).

Compute the Galois group of the polynomial \(x^{10}+x^5+1\) over \(\mathbb {Q}\).

Let \(F\) be a field and \(n\) a positive integer such that \(F\) has no nontrivial field extensions of degree less than \(n\). Let \(L=F[\alpha ]\) be an extension such that \(\alpha ^n\) is in \(F\). Prove that each element of \(L\) is a product of elements of the form \(a\alpha +b\), with \(a,b\) in \(F\).