Basics of Fields

Prove the following statements without using any results from the previous chapters.

1. $-0=0$.

2. $1^{-1}=1$.

Let $d$ be a square-free integer not equal to 1. Show that $\mathbb{Q}[\sqrt{d}]$ is a subfield of $\mathbb{C}$.

Find a ring without identity that is contained in a field.

Prove or disprove: $\mathbb{R}\cong\mathbb{C}$ as fields.

Show that $\mathbb{Z}_{3}[x]/\left\langle x^{2}+x+1\right\rangle$ is not a field.

Find an irreducible polynomial in $\mathbb{Z}_{2}[x]$. Hence construct a field $F$ of order 4. Find the prime subfield of $F$.

Consider the ring $R=\mathbb{Z}[i]/\left\langle 2-i\right\rangle$.

1. Show that $5\in\left\langle 2-i\right\rangle$ and $2+\left\langle 2-i\right\rangle=i+\left\langle 2-i\right\rangle$.

2. Show that $R=\{k+\left\langle 2-i\right\rangle\mid k\in\mathbb{Z}_{5}\}$.

3. Hence prove, via an isomorphism, that $R$ is a field with order 5.

Suppose $F$ is a subfield of $\mathbb{Q}$. Prove $F=\mathbb{Q}$.

Prove 1.21.

Let $F$ be a finite field with characteristic $p$, a prime.

1. Explain why $p$ divides $|F|$.

2. Suppose $q$ is another prime dividing $|F|$. Show there exist integers $\lambda$ and $\mu$ such that $(\lambda p+\mu q)x=x$ for all elements $x\in F$.

3. Deduce that the supposition in (ii) leads to a contradiction. Hence explain why $|F|$ must have order $p^{n}$, where $n$ is a positive integer.

Consider the field $F=\mathrm{Frac}(\mathbb{Z}_{p}[x])$.

1. Prove $F$ is infinite.(Hint: it would be easier to prove $\mathbb{Z}_{p}[x]$ is infinite.)

2. Prove $\mathrm{char}(F)\neq 0$.