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MAT102 Worksheet 7 Review

Q1

• Consider the set $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}:a,b\in \mathbb{Q}\}$, addition and multiplication are done as if it were in $\mathbb{R}$.

A

• Show that $\mathbb{Q}(\sqrt{2})$ is closed under addition and multiplication.
• Show that if $x,y\in \mathbb{Q}(\sqrt{2})$ then $x+y\in Q(\sqrt{2})$
  • Let us define $x=a+b\sqrt{2}$ and $y=c+d\sqrt{2}$
  • $x+y$
  • $a+b\sqrt{2}+c+d\sqrt{2}$
  • $a+c+b\sqrt{2}+d\sqrt{2}$
  • $a+c+\sqrt{2}(b+d)$
  • Let $x_0=a+c,y_0=b+d$
  • Notice we have $x_0+y_0\sqrt{2}$
  • Notice that $x_0\in \mathbb{Q},y_0\in \mathbb{Q}$
  • This is closed under addition
• Show that if $x,y\in \mathbb{Q}(\sqrt{2})$ then $xy\in Q(\sqrt{2})$
  • Let us define $x=a+b\sqrt{2}$ and $y=c+d\sqrt{2}$
  • $(a+b\sqrt{2})(c+d\sqrt{2})$
  • $ac+ad\sqrt{2}+bc\sqrt{2}+2bd$
  • $ac+2bd+ad\sqrt{2}+bc\sqrt{2}$
  • $ac+2bd+\sqrt{2}(ad+bc)$
  • let $x_0=ac+2bd,y_0=ad+bc$
  • Notice that $x_0,y_0\in \mathbb{Q}$
  • So we have $x_0+y_0\sqrt{2}$
  • This is closed under multiplication

B

• Show that $\mathbb{Q}(\sqrt{2})$ is a field.
  • Since $\mathbb{Q}(\sqrt{2})$ is a subset of $\mathbb{R}$. ($\mathbb{Q}\subseteq \mathbb{R}$) then we inherit a lot of axioms that don't need to be proved, like commutativity, associativity.
  • So let's prove inverses and identities
  • Identities:

    • Additive:

  • Inverses:
    • Additive:
      • We want to show that we have some $a\in \mathbb{Q}(\sqrt{2})$ where $-a\in \mathbb{Q}(\sqrt{2})$