1. Rings and Fields

We begin our exploration in the vibrant realm of abstract algebra, a field made prominent by mathematicians like Euler and Gauss, reaching its zenith in the mid to late 19th century with Kronecker's definition of an abelian group. During this time, Richard Dedekind contributed the concept of an algebraic structure known as a ring.

A ring is defined as a set ( R ) paired with two binary operations: addition ( + ) and multiplication ( cdot ), satisfying specific criteria:

1. ( R ) must form an abelian group under addition, meaning:
   • Addition is associative: ( (a + b) + c = a + (b + c) ) for any ( a, b, c in R ).
   • It is commutative: ( a + b = b + a ).
   • There exists an additive identity (denoted as 0) such that ( a + 0 = a ) for any ( a in R ).
   • Each element ( a in R ) has an additive inverse (denoted as ( -a )), satisfying ( a + (-a) = 0 ).
2. The multiplication operation is associative, and there exists a multiplicative identity (usually denoted as 1).
3. The multiplication operation is distributive over addition:
   • ( a cdot (b + c) = (a cdot b) + (a cdot c) ) and ( (a + b) cdot c = (a cdot c) + (b cdot c) ).

It is important to note that multiplication does not need to be commutative, although in many common rings, it is, leading to what are known as commutative rings.

Example 1.2: A familiar example of a ring is the set of integers ( mathbb{Z} ) under standard addition and multiplication, which is commutative. Similarly, the set of all polynomials with integer coefficients ( mathbb{Z}[x] ) forms a commutative ring. Conversely, the set of ( 2 times 2 ) matrices with real entries under standard addition and multiplication is an example of a non-commutative ring.

Rings may seem somewhat limited as structures due to the absence of a requirement for multiplicative inverses. Adding this condition leads to a more robust algebraic structure.

Definition 1.3: A field is a commutative ring ( F ) that possesses a multiplicative identity ( 1 ) distinct from the additive identity ( 0 ), and every element of ( F ) (except 0) has a multiplicative inverse.

Example 1.4: This distinction should be familiar, as it allows us to classify various number sets based on their algebraic properties. For instance, ( mathbb{Q} ) (the rationals) and ( mathbb{R} ) (the reals) are fields, whereas ( mathbb{Z} ) is not. Notably, fields are not necessarily infinite; the smallest field, known as the Galois Field ( GF[2] ), consists of just two elements: 0 and 1, with addition defined as XOR and multiplication as AND.

2. Fascinating Properties of Fields

Here are some intriguing facts about fields, mainly to steer us back to the context of Greek geometry:

Fact 2.1: Fields can be extended to larger fields, with the extension's degree corresponding to the dimension of the larger field as a vector space over the smaller one.

Example 2.2: The set of all complex numbers is a field extension of the real numbers, with an extension degree of 2, as all complex numbers can be expressed as pairs of real numbers.

Fact 2.3: In a sequence of finite degree field extensions, the degrees are multiplicative. If ( J, K, ) and ( L ) represent such a chain, with ( L ) having degree ( s ) over ( K ) and ( K ) degree ( t ) over ( J ), then ( L ) has degree ( st ) over ( J ).

Historically, fields and subfields were vital for understanding polynomial roots.

Fact 2.4.1: An element ( alpha ) of ( K ) is termed algebraic over ( J ) if it is a root of a polynomial in ( J[x] ). Otherwise, ( alpha ) is considered transcendental over ( J ).

Fact 2.4.2: A polynomial ( f ) in ( J[x] ) that has ( alpha ) as a root, and is of the lowest possible degree, is referred to as an irreducible polynomial over ( J ). The degree of this polynomial equals the degree of the field extension of ( K ) over ( J ).

Example 2.5: The square root of 2 is algebraic over ( mathbb{Q} ), with its irreducible polynomial being ( x^2 - 2 ). Hence, ( sqrt{2} ) lies within a field extension of degree 2 over ( mathbb{Q} ).

3. The Connection to Compass and Straight Edge Constructions

The limitations of compass and straight edge constructions are well established. Let's clarify these limitations.

Definition 3.1: Given two points in Cartesian space, the construction rules are defined as follows: 1. Any straight line connecting the two points is a valid construction. 2. Any circle centered at one point and intersecting another point is valid. 3. The intersection points of valid constructions are also valid.

These rules provide ample opportunity for construction. With a finite number of steps, one can create parallel lines, bisect angles, and find midpoints, practices familiar to many from school.

Definition 3.2: A number ( q ) is termed constructible if, starting from a unit line in Cartesian space, a line of length ( q ) can be drawn through a finite series of valid construction steps using a compass and straight edge.

Fact 3.3: It can be shown that the set of all constructible numbers forms a field under standard addition and multiplication, which implies that all rational numbers are constructible.

The facts regarding fields are particularly relevant here. A straight line formed through two points with coordinates in ( mathbb{Q} ) corresponds to a polynomial of degree 1 over ( mathbb{Q} ), while a circle formed with a center in ( mathbb{Q} ) and intersecting another point in ( mathbb{Q} ) corresponds to a polynomial of degree 2 over ( mathbb{Q} ). This leads us to two significant conclusions:

Fact 3.4: The intersection points of any two lines with coefficients in ( mathbb{Q} ) are also in ( mathbb{Q} ).

Fact 3.5: For a line and a circle, or between two circles with coefficients in ( mathbb{Q} ), their intersection points must reside within a field extension of degree at most 2 over ( mathbb{Q} ).

Fact 3.6: Consequently, any constructible number must lie within a field of degree ( 2^k ) over ( mathbb{Q} ) for some positive integer ( k ). Each construction step extends ( mathbb{Q} ) by degree 1 or 2, and since field extensions are multiplicative, this holds true.

(Note that a corollary of this is that all constructible numbers are algebraic over ( mathbb{Q} ).)

Now we can demonstrate the impossibility of what the Greeks were attempting:

Fact 3.7: Doubling the unit cube is impossible in a finite number of steps using a compass and straight edge. To double the unit cube, we need to establish that ( sqrt[3]{2} ) is constructible. However, as shown in the previous examples, ( sqrt[3]{2} ) is situated in a field extension of degree 3 over ( mathbb{Q} ), thus it cannot be constructible by Fact 3.6.

Fact 3.8: Squaring a circle with a unit radius is also impossible in a finite number of steps using a compass and straight edge. To square the circle, it is essential to prove that ( pi ) is constructible, which translates to demonstrating that ( pi ) is constructible. However, Lindemann established in 1882 that ( pi ) is transcendental over ( mathbb{Q} ), meaning no finite-degree polynomial with coefficients in ( mathbb{Q} ) can have ( pi ) as a root. Thus, ( pi ) lies within a field extension of infinite degree over ( mathbb{Q} ), leading us to conclude that ( pi ) cannot be constructible.

I find the relationship between abstract algebra and ancient Greek geometry to be both captivating and enlightening. I hope you do as well. If you’re interested in further exploration, I encourage you to tackle other impossible constructions. For instance, while we demonstrated how to bisect any angle, consider proving that it is impossible to trisect a 60-degree angle. (Hint: Look for an equation that expresses ( cos(3theta) ) in terms of ( cos(theta) ) and observe the implications of the logic presented in this article.)

What are your thoughts on the connection between algebra and classical geometry? Feel free to share your comments.