Prove that no order can be defined in the complex field that turns it into an ordered field. Hin

Prove that no order can be defined in the complex field that...

Key Concepts

Ordered Field

An ordered field is a field that is equipped with a total order which is compatible with the field operations. This means that if a > 0 and b > 0, then both a + b > 0 and a * b > 0, and crucially, the square of any nonzero element must be positive. This property is essential in distinguishing ordered fields from other algebraic structures.

Complex Field

The complex field extends the real numbers by including an element whose square is negative, namely the imaginary unit. This structure leads to elements that naturally defy the order properties required in an ordered field, such as the condition that the square of a nonzero element is always positive.

Square Property in Ordered Fields

In any ordered field, the square of each nonzero element must be strictly positive. This axiom is central to the incompatibility of the complex field with any order that would make it an ordered field, because in the complex field, an element exists whose square is -1, violating this requirement.

Transcript

00:02 We want to show that the complex number c does not satisfy the order properties.

00:08 That is, it is impossible to find a subset c plus of the complex numbers that satisfies properties similar to those stated for the real numbers.

00:24 So let's say we have here the properties of the real numbers regarding the order.

00:33 The set of real numbers contains a subset r plus satisfying the following properties.

00:42 First property, given any a in the real numbers, exactly one and only one of the following statements is true.

00:55 A is in r plus, a is zero, or negative a is in r plus.

01:03 The second property, if two numbers a and b are in r plus, then the sum of the numbers is also in r plus as well as the product of the numbers.

01:21 So we're going to prove that does not exceed a set, a subset of c plus of the nile complex number, satisfying these two properties.

01:30 For that, let's suppose that such a set exist.

01:34 So let's suppose that such a set c plus exists.

01:51 I'm not going to write again the properties.

01:54 We are clear that if we take any complex number, that is the translation of these two properties to the complex numbers, is putting here complex number and here c plus.

02:09 That is the subset c plus of c which we state that exists.

02:18 So going that, let's prove that is impossible.

02:23 That is supposing that this set c plus exists, we're going to get some contradiction.

02:33 Okay, so if that set exists, we know that if we take i, the complex number i, that is zero, in its real part and one it's in its imaginary part.

02:49 Then we get to have applying property 1 by property 1, i belongs to c plus or i is 0 or i or negative i belongs to c plus.

03:14 Remember that here when we talk about negative i, the sum, the product, we are talking about the product, the sum in the company.

03:23 Number...

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