# Exam Review



 Sections for the Exam: 

- 2.1-2.4 ($\mathbf R, \mathbf Q, \mathbf C$) and quadratic/cubic formulas
- 3.1-3.3 (Euclidean algorithm, roots, $\mathbf Z[x]$)
- 5.1 (Vector Spaces, toward field extensions)

Advice: for every definition or theorem, have an example and a counterexample in mind! E.g. when defining a ring, know by heart a good example of a set that *is* a ring and a set that is *not* a ring.



Note: some of this computations can be done by online calculators like WolframAlpha. Feel free to use these to check your work -- but also make sure you have tricks and techniques for doing it manually in an exam setting!



## Chapter 2



### Rings and fields



- Define what it means for a set $R$ to be a **ring**.
  - Define what it means for a subset $S \subseteq R$ to be a **subring**.
    - Let $A$ and $B$ be subrings of a ring $R$. Is $A \cap B$ a subring? Explain.

      - *Hint: true. How do you prove it?*
    - Let $A$ and $B$ be subrings of a ring $R$. Is $A \cup B$ a subring? Explain.

      - *Hint: false. Consider $a+b$ where $a\in A, b\in B$.*

    - Let $A$ and $B$ be subrings of a ring $R$. Is $A+B := \{ a+b\mid a\in A, b\in B\}$ a subring? Explain.

      - *Hint: false, consider $(a_1 + b_1)(a_2 + b_2)$*.
  - Prove that $M_2(\mathbf Z)$, the set of 2 by 2 matrices with integer coefficients, is a ring.
  - Prove that $GL_2(\mathbf{Z}) := \left\{ M\in M_2(\mathbf Z) \mid \det M \neq 0 \right\}$ is not a ring.
- Define an **integral domain**.
  - Define a **zero divisor**.
  - Prove that $\mathbf{Z}/5\mathbf{Z}$ is an integral domain.
  - Prove that $\mathbf{Z}/6\mathbf{Z}$ is not an integral domain.
  - Prove that if $R$ is an integral domain, then its polynomial ring $R[x]$ is also an integral domain. (See p. 84)
  - Prove that $\mathbf{Z}/n\mathbf{Z}$ is an integral domain $\iff n$ is prime.
  - Let $R$ be an integral domain and assume that $a b=a c$, where $a, b, c \in R$ and $a \neq 0$. Prove that $b=c$.
- Define a **field**.
  - Define a **unit** in a ring.
  - Prove that if $R$ is a field then $R$ is an integral domain.
  - Prove that $\mathbf{Z}/5\mathbf{Z}$ is a field by showing the multiplicative inverse of every element exists in the ring.
  - Prove that $\mathbf{Z}/6\mathbf{Z}$ is not a field by finding an element without a multiplicative inverse.
  - Let $R = \mathbf{Z}/m\mathbf{Z}$ for $m$ arbitrary. When is an element $\bar a\in R$ a unit?
- Define an **ordered field** (see p. 47).
  - Prove that $\mathbf{Q}$ is an ordered field (see p.48)
- Prove that $\sqrt 2$ is irrational.
- Define the **least upper bound** of a set (see p. 52)
- Prove that $\mathbf{Q}[\sqrt 2] := \left\{ a_0 \cdot 1 + a_1 \cdot \sqrt{2} \mid a_0, a_1\in \mathbf{Q} \right\}$ is a field (see p.54).
  *Hint: it suffices to find the multiplicative inverse of $a_0 + a_1\sqrt{2}$ and show it is still an element of $\mathbf{Q}[\sqrt 2]$*.
  - Prove that $\mathbf{Q}[2^{1\over 3}]$ is a field.
    *Hint: find a basis and write it as a set as above.*
  - Prove that $\mathbf{Q}[\sqrt d]$ is a field for any $d\in\mathbf{Z}$.

- Write down all of the elements in $$R=\left\{\left(\begin{array}{ll}a & b \\ 0 & a\end{array}\right) \mid a, b \in \mathbf{Z}/2\mathbf Z\right\} \subset M_2\left(\mathbf{Z}/2\mathbf{Z}\right)$$. 
  Is $R$ commutative? What are the additive and multiplicative inverses of all elements? Is $R$ an integral domain? Is $R$ a field?

### Complex numbers and field extensions

- Let $R$ be a ring and $f\in R[x]$ a polynomial. Define what it means for $f$ to be **irreducible** in $R[x]$.
  - State the **rational root test**.
  - State **Eisenstein's criterion**.
  - Let $f\in \mathbf{Z}[x]$ be a polynomial with integer coefficients. How does the irreducibility of $f\in \mathbf{Z}[x]$ relate to the reducibility of $\bar f\in {\mathbf{Z} \over p\mathbf{Z}}[x]$? How can you use this to prove or disprove irreducibility of a polynomial?
    *Hint: there is a statement like "if $f$ is reducible over $\mathbf{Z}$, then $\bar f$ is ??? when reduced mod $p$".*
  - Prove that $f(x) = x^2-2$ is irreducible in $\mathbf{Q}[x]$.
  - Prove that if $f\in R[x]$ has a root $\alpha\in R$, then $f$ is reducible in $R[x]$.
  - Prove that $x^4-5x^2+6$ has no rational roots $\alpha\in \mathbf{Q}$, but is still reducible in $\mathbf{Q}[x]$.
  - Show that $p(x)=x^3+x+1$ is irreducible in ${\mathbf Z \over 2\mathbf Z}[x]$.
  - Let $\mathbf{F}=\mathbf{Z}/2\mathbf{Z}$ and factor the polynomial $x^8+x \in \mathbf{F}[x]$ into irreducible factors.
  - Prove that $h(x)=x^4+2 x^2+5 x+1$ is irreducible over $\mathbf{Q}$ by reducing mod $p$ for some prime $p$.
  - Prove that the polynomial $f(x)=1+x+x^3+x^4$ is reducible over *any* field $\mathbf{F}$.
- Let $R$ be a ring and $f, g\in R[x]$ be two polynomials. Define what the **greatest common divisor (GCD)** of $f$ and $g$ is, usually written $d(x) = \gcd(f(x), g(x))$.
  - Suppose that $f(x)=x^{99}-5 x^{49}+1 \in \mathbf{Q}[x]$. Compute the remainder when $f(x)$ is divided by
    - $g_1(x) = x$;
    - $g_2(x) = x-1$;
    - $g_3(x) = x+1$.

- State **Euler's formula for complex numbers**.
  *Hint: this is the one that converts $e^{i\theta}$ to cosines and sines.*

  - Convert $5+5i$ to polar coordinates  
  - $7e^{i\pi\over 3}$ to rectangular coordinates.
  - Let $z = re^{i\theta}$ and $w = \rho e^{i\phi}$, and prove that $z w=r \rho(\cos (\theta+\phi)+i \sin (\theta+\phi))$.
  - Prove de Moivre's theorem:
    $(\cos(x) +i \sin(x))^n=\cos(n x)+ i \sin (n x)$

- Practice factoring:

  - Factor $f(x) = x^n-1$ into irreducible polynomials.
  - Factor $f(x) = x^n-a$.
    *Hint: you'll need a term like $a^{1\over n}$. Try some concrete examples like $a=2, 3,4,\cdots$.*
  - Factor $f(x) = x^n+1$ 
  - Factor $f(x) = x^n + a$

- Perform the division algorithm to compute ${x^3+2 \over 2x^2 + x + 1}= (3x+1) + {x+1\over 2x^2+x+1}$. (See p.85)

- Define the gcd of two polynomials. How do you compute it?

- Let $d(x) = \gcd(x^3-8, x^2-x-2)$ and show $d(x) = x-2$.

  Find polynomials $a(x), b(x) \in \mathbf{Q}[x]$ such that 
  $$
  d(x) = a(x)f(x) + b(x) g(x)
  $$
  

  and by a computation, show that 
  $$
  d(x) = {1\over 3}f(x) + (-{1 \over 3 })(x+1)g(x)
  $$
  

- Find the greatest common divisor of the following polynomials over $\mathbf{Q}$.
  - $x^2+x-2$ and $x^5-x^4-10 x^3+10 x^2+9 x-9$
  - $x^2+1$ and $x^6+x^3+x+1$.

## Splitting Fields and Extensions



- Let $f\in \mathbf{Q}[x]$ and define what the **splitting field** of $f$ is.
  - Find the splitting field of $f(x) = x^3-1$.
  - Find the splitting field of $f(x) = (x^2-2)(x^2+1)$.
  - Find the splitting field of $f(x) = x^2+x+1$.
- Let $\alpha \in \mathbf{C}$ and define what the **minimal polynomial** of $\alpha$ is.
  - How does the degree of the minimal polynomial of $\alpha$ relate to the dimension of $\mathbf{Q}[\alpha]$ as a vector space over $\mathbf{Q}$?

- Show $\mathbf{Q}[\sqrt{3} + i] = \mathbf{Q}[\sqrt 3, i]$.
- For any $a, b\in \mathbf{Z}$ with $a,b>0$, prove that $\mathbf{Q}[\sqrt a, \sqrt b] = \mathbf{Q}[\sqrt a + \sqrt b]$.
  *Hint: you may assume $a\neq b$; prove subset containment both ways.*

- Find a *power basis* for $K = \mathbf{Q}[\sqrt 3, \sqrt 7]$, which is defined to be an element $\alpha\in K$ such that the powers $\left\{\alpha^0, \alpha^1,\cdots, \alpha^n \right\}$ form a basis for $K$ as a vector space over $\mathbf{Q}$.
  *Hint: $K = \mathbf{Q}[\alpha]$ for $\alpha = \sqrt 3 + \sqrt 7$ by a previous problem. What is the minimal polynomial of $\alpha$? What is the dimension of $K$?*