06-240/Homework Assignment 2

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Read sections 1.1 through 1.3 in our textbook, and solve the following problems:

• Note that the numbers [math]\displaystyle{ 1^6-1=0 }[/math], [math]\displaystyle{ 2^6-1=63 }[/math], [math]\displaystyle{ 3^6-1=728 }[/math], [math]\displaystyle{ 4^6-1=4,095 }[/math], [math]\displaystyle{ 5^6-1=15,624 }[/math] and [math]\displaystyle{ 6^6-1=117,648 }[/math] are all divisible by [math]\displaystyle{ 7 }[/math]. The following four part exercise explains that this is not a coincidence. But first, let [math]\displaystyle{ p }[/math] be some odd prime number and let [math]\displaystyle{ {\mathbb F}_p }[/math] be the field with p elements as defined in class.
  1. Prove that the product [math]\displaystyle{ b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1) }[/math] is a non-zero element of [math]\displaystyle{ {\mathbb F}_p }[/math].
  2. Let [math]\displaystyle{ a }[/math] be a non-zero element of [math]\displaystyle{ {\mathbb F}_p }[/math]. Prove that the sets [math]\displaystyle{ \{1,2,\ldots,(p-1)\} }[/math] and [math]\displaystyle{ \{1a,2a,\ldots,(p-1)a\} }[/math] are the same (though their elements may be listed here in a different order).
  3. With [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] as in the previous two parts, show that [math]\displaystyle{ ba^{p-1}=b }[/math] in [math]\displaystyle{ {\mathbb F}_p }[/math], and therefore [math]\displaystyle{ a^{p-1}=1 }[/math] in [math]\displaystyle{ {\mathbb F}_p }[/math].
  4. How does this explain the fact that [math]\displaystyle{ 4^6-1 }[/math] is divisible by [math]\displaystyle{ 7 }[/math]?

This assignment is due at the tutorials on Thursday September 28. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.