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Exhibit A
The condition (Na, Nb) \equiv (a, b) \mbox{ for all } N\neq 0
Exhibit B
If we represent a complex number a+bi by the ordered pair (a,b), we get the required properties:
(a+bi) + (c+di) = (a+c) + (b+d)i \iff (a,b) + (c,d) = (a+c, b+d)
(a+bi) \times (c+di) = (ac-bd) + (ad+bc)i \iff (a,b) \times (c,d) = (ac-bd, ad+bc)
Exhibit C
These formally define addition and multiplication over the natural numbers. Can you see how the familiar properties we're used to follow from them?
The first implies k+1 = 1+k, i.e. addition is commutative.
The second implies k+(1+n) = 1+(k+n), i.e. addition is associative.
The third implies k\times 1 = k, i.e. 1 is the multiplicative identity.
The fourth implies k\times(1+n) = k+(k\times n), that mulitplication is distributative over addition.
This is a rigorous treatment of a very familiar concept. For more information on this subject, you could start by reading this Wikipedia article.
Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.
This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.
Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.