Sample questions

1.

Groups

(a)

Isomorphism Theorems. Let M and N be normal subgroups of G such that G=MN. Prove that $G/(M\cap N)$ is isomorphic to $(G/M)\times (G/N)$.

(b)

Sylow theorem. Prove that if |G|=132 then G is not simple.

(c)

Groups actions. Suppose |G|=p^a, where p is a prime. Prove that every subgroup of index p is normal in G.

(d)

Cauchy-Frobenious-Burnside lemma. If there are q colors available, prove that there are (qⁿ²+2q^[(n2+3)/4]+q^[(n2+1)/2])/4 distinct $n\times n$ colored chessboads.

(e)

Linear groups. Let K be a field. Prove that GL(n,K) is a semidirect product of SL(n,K) by $K^{\times}=K-\{0\}$.

(f)

Split extension. Construct a non-abelian group of order 75.

(g)

Solvable groups. Suppose |G|=pq, where p and q are primes. Prove that G is solvable.

(h)

Finitely generated abelian groups. Let $G=Z_{60}\times Z_{45}\times Z_{12}\ times Z_{36}$. Find the number of elements of order 2 in G.

2.

Rings.

(a)

Polynomial rings. Let f(x) be a polynomial in F[x], where Fis a field. Prove that F[x]/(f(x)) is a field if and only if f(x) is irreducible.

(b)

Euclidean domains. Prove that the quotient ring Z[i]/I is finite for any nonzero ideal I of Z[i]. (Z[i] is the ring of Gausian integers).

(c)

Principal ideal domains. Let $I=(2,1+\sqrt{-5})$ be an ideal of $Z[\sqrt{-5 }]$. Prove that I is not a principal ideal of $Z[\sqrt{-5 }]$.

(d)

Unique factorization domains. Determine all the representations of the inte ger $2130797=17^2\cdot73\cdot101$ as a sum of two squares.

3.

Fields.

(a)

Galois Theory and construction of Galois groups. Determine the Galois group of x⁴+4x-1.

(b)

Finite fields. Write out the multiplication table for F₄ (the field of 4 elements).

(c)

Algebraic extensions. Prove that [ $Q(\sqrt{2}+\sqrt{3}):Q]=4$, where Q is the field of rational numbers.