Ph.D. Preliminary Examination in Algebra

June 9, 2000

1. Let $X$ be any set, $F$ any field, and $F^X$ the set of maps from $X$ to $F$. $F^X$ is endowed with the structure of a vector space over $F$ using “pointwise” addition and multiplication by scalars. Prove that a finite sequence $f_1,f_2,\dots ,f_n$ of elements of $F^X$ is linearly independent if and only if there is a finite sequence of elements $x_1,x_2,\dots ,x_n$ in $X$ for which the $n\times n$ determinant $\det \left(f_i\left(x_j\right)\right)$ is non-zero.

2. Find a complete set of representatives for the isomorphism classes of finite abelian groups of order $1001$.

3. Let $K$ be the splitting field over the field $Q$ of rational numbers of the polynomial $$f\left(x\right)=x^5-x^4+x^3-x^2+x-1\text{.}$$

   1. What are the possible values for the minimum degree among the irreducible factors of a polynomial of degree $5$ ?

   2. Write $f$ as the product of factors irreducible over $R$.

   3. Write $f$ as the product of factors irreducible over $Q$.

   4. What is the degree of $K$ over $Q$ ?

   5. What is the Galois group of $K$ over $Q$ ?

4. Show that if two square matrices of the same finite size over a field are similar in a larger field then they must be similar in the original field.

5. Let $F$ be a field, and let $A$ be the quotient ring $$A=F[t,x,y,z]/\left(tz-xy\right)F[t,x,y,z]$$ where $t,x,y,z$ are independent transcendentals over $F$.

   1. Show that $A$ has no zero divisors.

   2. Explain briefly why $A$ is Noetherian.

   3. Is $A$ a unique factorization domain? (Either prove that it is or exhibit an example of something that does not factor uniquely according to the usual criteria for such uniqueness.)

6. Let $E$ be a finite extension of a field $F$.

   1. Outline an argument for showing that if $F$ is a finite field, then $E$ is a cyclic Galois extension of $F$.

   2. Provide an example where $F$ is a field of characteristic $5$ and $E$ is an extension of $F$ of degree $5$ that is not a Galois extension of $F$.

   3. For any given field $K$ explain how to obtain an extension $F$ of $K$ and a finite extension $E$ of $F$ for which $E$ is a Galois extension of $F$ with Galois group isomorphic to the symmetric group $S_n$ (consisting of the permutations of $n$ objects).

7. Let $F_3$ denote the field of $3$ elements.

   1. What is the cardinality of $2$-dimensional Cartesian space $F_3\times F_3$ over $F_3$ ?

   2. Let $N$ denote cardinality of the group ${\mathrm{GL}}_2\left(F_3\right)$ of linear automorphisms of $F_3\times F_3$.
      Compute $N$.

   3. Observe that the multiplicative group ${F_3}^{*}$ is the unique group of order $2$ and furthermore that:

      1. Multiplication by invertible scalars gives rise to a homomorphism $\varphi$ from ${F_3}^{*}$ to ${\mathrm{GL}}_2\left(F_3\right)$.

      2. The determinant gives rise to a homomorphism $\psi$ from ${\mathrm{GL}}_2\left(F_3\right)$ to ${F_3}^{*}$

      Explain why the kernel of $\psi$ and the cokernel of $\varphi$ both have the same cardinality.

   4. Is the kernel of $\psi$ isomorphic to the cokernel of $\varphi$ ?

8. Prove over any commutative ring (with $1$) that two isomorphic free modules of finite rank must have the same rank.