Math 104 Assignment 2

1. Translation of suprema

   Suppose that $(F, \preceq)$ is an ordered field, that $H \subseteq F$, and that $a \in F$. Suppose further that $\sup H = t \in F$. Finally, let $H+a = \set{x + a \mid x \in H} \subseteq F$. Prove that $H+a$ has a supremum in $F$, and that $\sup(H+a) = \sup(H)+a$.

2. Some useful facts

   Prove the following useful facts.

   1. If $T \subseteq \Z$ is non-empty and bounded above in $\R$, then $\sup T \in T$. (In particular, $\sup T \in \Z$.)

   2. Suppose that $\emptyset \subsetneq E_1 \subseteq E_2 \subseteq \R$. If $E_2$ is bounded above, then $\sup E_1$ exists (in $\R$) and $\sup E_1 \leq \sup E_2$.

      (Note: the symbol "$\subsetneq$" means "is a proper subset of", so $A \subsetneq B$ means $A \subseteq B$ and $A \neq B$. Thus $\emptyset \subsetneq E_1$ means "$E_1$ is non-empty". In contrast, the symbol $\nsubseteq$ means "is not a subset of". The symbol "$\subset$" is inconsistently used, usually meaning $\subsetneq$ but sometimes meaning $\subseteq$, depending on the author.)

   3. If $x, y \in \R$ with $0 \leq x \leq y$ then $x^2 \leq y^2$. Moreover, assuming that $x^{1/2}, y^{1/2}$ exist in $\R$ with $0 \leq x^{1/2}, y^{1/2}$, prove that $x^{1/2}\leq y^{1/2}$.

   4. If $S$ is an ordered set, $E, F \subseteq S$ have suprema in $S$, and they have the property that for every $e \in E$ there is $f \in F$ with $e \preceq f$, then $\sup E \preceq \sup F$.

3. Some facts about suprema

   1. Let $(S, \preceq)$ be an ordered set, and $t \in S$. Prove that $\sup\set{s \in S \mid s \preceq t} = t$.
   2. Show by example that Part A need not hold if the $\preceq$ in the definition of the set is replaced by $\prec$.
   3. Show that, nonetheless, if $x \in \R$ then $$\sup\set{q \in \Q \mid q \lt x} = x.$$ (Note that, as a result, every real number is the supremum of some set of rational numbers.)

4. Suprema in the rationals are suprema in the reals

   1. Suppose that $E \subset \Q$ has least upper bound $t \in \Q$. Show that $t$ is also the least upper bound of $E \subset \R$.
   2. Prove that $\set{q \in \Q \mid q^2 \lt 2}$ does not have a least upper bound in $\Q$ (and so $\Q$ indeed does not have the Least Upper Bound Property).

5. Properties of irrational numbers

   1. Prove that if $q \in \Q$ and $t \in \R\setminus \Q$, then $q + t \in \R\setminus\Q$.
   2. Prove that if $q \in \Q$ and $t \in \R\setminus \Q$, then $qt \in \set{0}\cup(\R\setminus\Q)$.
   3. Prove that if $x, y \in \R$ with $x \lt y$, there is some $t \in \R\setminus\Q$ with $x \lt t \lt y$.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment. They are coloured by approximate difficulty: easy/medium/hard.

• Variations on a theme

  1. Suppose that $(\F, \leq)$ is an ordered field, $E \subseteq \F$, and $\alpha \in \F$. Prove that $t \in \F$ is a lower bound for $E$ if and only if $t + \alpha$ is a lower bound for $E + \alpha$. Consequently, prove that $\inf E$ exists if and only if $\inf(E+\alpha)$ exists, in which case $(\inf E)+\alpha = \inf(E+\alpha)$.
  2. Formulate and prove a similar statement about upper bounds and suprema of sets, when multiplied by some $t \in \F$ with $t \gt 0$.
  3. Formulate and prove a similar statement when $t \lt 0$.
  4. Prove that $\Z$ is not bounded below in $\R$.
  5. Prove that $\N$ is not bounded above in $\R$.
• Suppose that $x, y \in \R$ with $0 \leq x \lt y$, and $n, m \in \N$. Prove that $x^n \leq y^n$ if and only if $x^m \leq y^m$, and $x^{\frac1n} \leq y^{\frac1n}$ if and only if $x^{\frac1m} \leq y^{\frac1m}$.

• Ordered fields

  Suppose that $\F$ is a field. If $p \in \N$ is such that \[0_\F = \sum_{i=1}^p 1_\F = \underbrace{1_\F + 1_\F + \ldots + 1_\F}_{p \text{ times}},\] and $p$ is minimal with this property, then $\F$ is said to be "of characteristic $p$". If $\F$ is not of characteristic $p$ for any $p \in \N$, we say that $\F$ is of characteristic zero.

  1. Suppose that $p \in \N$. Prove that a field of characteristic $p$ exists if and only if $p$ is prime. (Hint: show that a field of characteristic $p$ must include $\Z / p\Z$, which has non-invertible elements whenever $p$ is not prime.)
  2. Prove that $\F$ is of characteristic zero if and only if there is a field homomorphism \[\varphi : \Q \longrightarrow \F.\] That is, a map $\varphi$ so that $\varphi(1) = 1_F$, $\varphi(0) = 0_F$, and for all $a, b \in \Q$, $\varphi(ab) = \varphi(a)\varphi(b)$ and $\varphi(a+b) = \varphi(a) + \varphi(b)$. Moreover, prove that $\varphi$ is unique and injective.
  3. Prove that any field homomorphism is injective (i.e., one-to-one).

  Now, let us suppose that $(\F, \preceq)$ is an ordered field.

  4. Prove that $\F$ cannot be bounded above.
  5. Prove that $\inf\set{x \in \F \mid x \gt 0} = 0$.
  6. Prove that $\F$ is of characteristic zero. Notice that this implies that $\F$ is infinite.
  7. Let $\varphi$ be the embedding from Part B. Prove that $\varphi$ preserves order: that is, for any $a, b \in \Q$, $a \leq b$ if and only if $\varphi(a) \preceq \varphi(b)$.
  8. Prove that if $R_1$ and $R_2$ are ordered fields with the Least Upper Bound Property, then there is an order-preserving field isomorphism $R_1 \to R_2$.

  Let $F = \Q[X] / \paren{X^2-2}$; this is the field of polynomials with rational coefficients in a single indeterminate $X$, modulo the ideal generated by $X^2-2$. Every element of $F$ is of the form $s + tX$, with addition defined component-wise and multiplication defined by \[(s_1+t_1X)\cdot_F(s_2+t_2X) = (s_1s_2+2t_1t_2) + (s_1t_2+t_1s_2)X.\]

  9. Verify that $F$ is a field. (This one is "easy" in that it's straightforward, but probably not short.)
  10. Find all field automorphisms of $F$ which fix $\Q$; that is, all isomorphisms $\varphi: F \to F$ so that $\varphi(q) = q$ for all $q \in \Q$.
  11. Show that there are precisely two ways of making $F$ an ordered field, one in which $X \lt 0$ and one in which $0 \lt X$.
  12. Let $N \in \N$; show that there is a field with $N$ elements if and only if there are $n, p \in \N$ with $p$ a prime so that $N = p^n$.
  13. Suppose that $F_1, F_2$ are finite fields with the same number of elements. Prove that they are isomorphic.