189-245A: Algebra 1

Assignment 1

Due: Monday, September 15.

1. Use Cardano's formula to solve the following cubic equations. In each case say how many real solutions there are and list all such solutions when there are more than one. (You are advised to use a calculator to check that the expressions you've written down are indeed solutions to the equation at hand.)

a. $x^3+3x+1$

b. $x^3-3x+1$. (In this case, give a closed form expression for the solution(s) of the equation, in terms of $\cos(2\pi/9)$ and $\sin(2\pi/9)$.)

2. Let $R$ be a finite ring without zero divisors (i.e., non-zero elements $a,b\in R$ satisfying $ab=0$). Show that every non-zero element of $R$ is invertible. Use this to conclude that ${\mathbb Z}/p{\mathbb Z}$ is a field.

3. Exercise (9) on page 29 of the on-line notes.

4. Exercise (26) of page 31-32 of the on-line notes.

5. Given any set $A$, show that the cardinality of $A$ is strictly less than that of its power set.

6. Let $X$ be a set, and let ${\cal F}(X)$ be the set of all functions from $X$ to itself. This set is equipped with a natural binary operation $(f,g) \mapsto fg$ , given by the composition of functions.

a. Show that $f(gh) = (fg)h $ for all $f$, $g$, $h$ in ${\cal F}(X)$. (In other words, the operation of composition of funcions is always {\em associative}.)

b. Show, by providing an example, that $fg$ need not be equal to $gf$, i.e., that composition of functions {\em need not be commutative}.

7. Show that there are infinitely many primes of the form $3n+2$ and of the form $4n+3$, with $n\ge 1$.

8. Without resorting to a calculator or computer, write the complex number $(1+i)^{83}$ in the form $a+bi$ where $a$ and $b$ are real numbers. Explain how you proceeded.

9. Using the Euclidean algorithm compute the gcd of $123654$ and $321456$. Show the steps in your calculation.

10. Using induction, show that the addition law in $\mathbf N$ is associative directly from the axioms defining addition in $\mathbf N$ in terms of the successor function.