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Determine, up to isomorphism, the groups of order 44.

Prove that there is no simple group of order 192.

Let \(R=\mathbb {Q}[x,y]\) and let \(I=\langle x,y\rangle \). Compute the vector space rank of \(\operatorname {Ext}^{1}_{R}(I,R)\) over \(\mathbb {Q}\).

Let \(M\) be a \(3\times 3\) complex matrix with \(M^6=M^4\) and \(M^4+M^2=2M^3\). Determine possible Jordan forms of \(M\).

1. List the prime ideals of the ring \(R=\mathbb {Z}[x,y]/\langle 3+x, y(y-1)+x^2, x\rangle \).
2. Give an example of an integral domain with exactly 2 prime ideals.

Suppose that \(R=\mathbb {Z}[i]\) is the Gaussian integers (here \(i\) is a square root of \(-1\)). Let \(M\) and \(N\) be finitely generated \(R\)-modules such that \[ M\oplus R^{\oplus 2}\oplus R/\langle 2\rangle \cong N\oplus R\oplus R/\langle 1+i\rangle \oplus R\oplus R/\langle 1-i\rangle . \] Is it true that \(M\cong N\)?

Prove that \(\mathbb {F}=\mathbb {Z}[t]/\langle 3, t^3-t^2+1\rangle \) is a field. Find the number of solutions of \(x^{13}+1=0\) in \(\mathbb {F}\), and also the number of solutions of \(x^{13}-1=0\) in \(\mathbb {F}\).

Compute the Galois group of \(x^4-2\) over \(\mathbb {Q}\).

Let \(E\) be the splitting field of \(f(x)=x^{14}+1\) over \(\mathbb {F}_2\), and let \(K\) be the splitting field \(g(x)=x^{21}+1\) over \(\mathbb {F}_2\). Prove that \(K\) contains an isomorphic copy of \(E\), and compute the extension degree of \(K\) over \(E\).

Let \(f(x)\) be a degree 5 polynomial in \(\mathbb {Q}[x]\) that is not solvable by radicals. Let \(L\) be its splitting field over \(\mathbb {Q}\).

1. Prove that there is at most one field \(K\) with \(\mathbb {Q}\subset K\subset L\) and \([K:\mathbb {Q}]=2\).
2. If \(\alpha ,\beta \) are irrational elements in \(L\) such that \(\alpha ^2\) and \(\beta ^2\) are rational, prove that \(\alpha \beta \) is rational.