Complex Numbers As A Field Extension

1.0 Introduction

This is some well-established mathematics. I do not know of any use in economics.

2.0 The Field of Real Numbers

I start by taking real numbers R, with binary operations for addition and multiplication, as known.

3.0 A Ring of Polynomials

Consider polynomials with coefficients taken from the reals as a formal object:

R[X] = { a_n Xⁿ + ... + a₁ X + a₀ | n ≥ 0,

coefficients from the reals. }

The symbol X is known as an indeterminate. One does not consider it here as a member of some set. Polynomial addition is defined to yield another polynomial, with addition of coefficients in the reals. Polynomial multiplication is also defined as usual.

3.1 Evaluation Homomorphisms

Still, one would like to talk about evaluating polynomials. For every real number r, there exists an evaluation homomorphism φ_r( ) that maps R[X] into the reals. This homomorphism is defined by:

φ_r( a_n Xⁿ + ... + a₁ X + a₀ ) = a_n rⁿ + ... + a₁ r + a₀

Addition and multiplication on the right-hand side above is performed in the reals. The map is a homomorphism because it preserves addition and multiplication:

φ_r( f(X) + g(X) ) = φ_r( g(X) ) + φ_r( g(X) )

φ_r( f(X) g(X) ) = φ_r( g(X) ) φ_r( g(X) )

In words, it does not matter, in evaluating the sum or product of polynomials if:

• You perform the operation in the polynomial ring first and then evaluate the sum, or
• You evaluate the polynomials and then sum or multiply in the reals.

A homomorphism that is one-to-one is an isomorphism. These evaluation homomorphisms are not isomorphisms since more than one polynomial may be evaluated to have the same value. An example follows:

φ₂(X²) = φ₂(X + 2) = 4

3.2 A Polynomial Without a Multiplicative Inverse

The constant polynomials 0 and 1 are the additive and multiplicative identities for polynomial addition and multiplication, respectively. These identities are distinct.

Not all polynomials have a multiplicative inverse. A simple example is the polynomial X. Suppose f(X) were a polynomial in R[X] that was the multiplicative inverse of X. Then:

X f(X) = 1

Consider the evaluation homomorphism for the additive identity in the reals.

1 = φ₀(X f(X)) = φ₀(X) φ₀(f(X)) = 0 φ₀(f(X)) = 0

So the non-existence of a multiplicative inverse for X is proven by a proof by contradiction.

It has been demonstrated that R[X] cannot be a field, since not every non-zero element has a multiplicative inverse. I believe it is actually an integral domain. Just as the field of rational numbers can be constructed as equivalence classes of ordered pairs of integers, a field of rational polynomials with real coefficients can be constructed. I do not pursue this construction here.

4.0 Polynomial Addition and Multiplication Modulo p(X)

One can define the quotient q(X) and remainder r(X) for any polynomials f(X) and g(X) in R[X]:

f(X) = q(X) g(X) + r(X)

where r(X) is of degree less than g(X). Since the reals are a field, the quotient and remainder are unique.

The above theorem allows one to define polynomial addition and multiplication modulo p(X). In particular, consider:

p(X) = X² + 1

p(X) is irreducible. There do not exist non-constant polynomials f(X) and g(X) in R[X such that:

p(X) = f(X) g(X)

I now define the set C of polynomials

C = { r(X) | there exists a f(X) in R[X]

such that r(X) = f(X) mod p(X)}

All polynomials in C are at most of degree one.

4.1 C as a Two-Dimensional Vector Space

Each element of C can be expressed as a linear combination of the elements of the basis {X, 1}:

C = { (a₁, a₀) | a₁ X + a₀ is in R[X]}

4.2 C as a Field Extension of the Reals

Consider C with addition and multiplication defined modulo p(X). I claim this is a field. Consider:

f(X) = a₁ X + a₀

g(X) = (-a₁/(a₀² + a₁²)) X + a₀/(a₀² + a₁²)

Their product in R[X] is:

f(X) g(X) = ((-a₁²/(a₀² + a₁²)) X² + a₀²/(a₀² + a₁²))

The quotient and remainder are found from:

f(X) g(X) = p(X) (-a₁²/(a₀² + a₁²)) + 1

Or:

(f(X) g(X)) mod p(X) = 1

So every non-zero element of C has a multiplicative inverse.

The set of constant polynomials in C is isomorphic to the reals. Thus, C extends the reals in a precise sense.

4.3 Evaluation Homomorphisms in C

For every a₁ X + a₀ in C, one can define an evaluation homomorphism φ_{(a1 X + a0)}( ) that maps R[X] into C. For every constant polynomial in C, this evaluation homomorphism yields the same answer as the corresponding evaluation homomorphism in Section 3.1.

As an example of this evaluation homomorphism, consider:

φ_X(p(X)) = φ_X(X² + 1)

Or:

φ_X(p(X)) = φ_X(X²) + φ_X(X¹)

Or:

φ_X(p(X)) = (XX) mod p(X) + 1

With addition and multiplication in R[X]:

X² = p(X) - 1

Thus:

φ_X(p(X)) = -1 + 1 = 0

In other words, the polynomial X in the field extension C is the square root of -1.

5.0 Summary

The above has extended the field of reals to the field of complex numbers. This field extension contains a zero for the equation:

p(X) = X² + 1 = 0

Furthermore:

• The imaginary numbers are polynomials of degree one and no constant term, with addition and multiplication defined modulo p(X).
• The real numbers are isomorphic to constant polynomials, with addition and multiplication defined modulo p(X).

That is, the extension field C is the field of complex numbers. The complex numbers are only defined up to isomorphism. But their existence is constructed here, not postulated.

6.0 Other Field Extensions

One need not begin this exposition with polynomials with coefficients from the real numbers. Coefficients can be drawn from other fields.

For example, consider the set {0, 1, 2, ..., p - 1}, with addition and multiplication defined modulo p, and p prime. This set with these operations is a field. Let p(X) be, as above, an irreducible polynomial in the ring of polynomials with coefficients in the set. Suppose p(X) is of degree n. Then the field extension is the Galois Field, GF(pⁿ). The set of elements of GF(2ⁿ) - {0}, with multiplication, is a cyclic group. GF(2ⁿ) has application in the Advanced Encryption System (AES) and in Reed-Solomon error correction codes. (The latter has something to do with how checkout scanners work in your neighborhood supermarket.)

On the other hand, consider the field of rational numbers with addition and subtraction defined as usual. There are at most a countably infinite number of polynomials with rational coefficients. An irreducible polynomial leads to an extension field for use in constructing real numbers. But this construction leaves out an uncountably infinite number of real numbers, namely the transcendental real numbers. A real number is algebraic if it is the root of some polynomial with rational coefficients. The real numbers, including transcendentals, can be constructed, instead, as Dedekind cuts or as equivalence classes of Cauchy-convergent sequences of rational numbers. (Cauchy often comes across as a villain in accounts of Galois and Abel's short lives.)

Finally, consider polynomials with coefficients drawn from the field of complex numbers. (Since, under the above construction, a complex number is, in some sense, a first-degree polynomial with real coefficients, this may be a somewhat confusing construction to think about.) Suppose one defines polynomial addition and multiplication modulo p(X), where p(X) is a first degree polynomial in the ring C[X]. Then one obtains a field "extension" isomorphic to the field of complex numbers.

To find a bigger field extension, one needs to find an irreducible polynomial of at least degree two in C[X]. But no such polynomial exists. Proof: Abel was something else, wasn't he?

Lack Of Rigor In Mas-Colell, Whinston, And Green

Whether or not the government should intervene in the economy is a false choice. Government and the economy are not two separate and non-intertwined entities.

The standard introductory graduate microeconomics textbook now current was written by Mas-Colell, Whinston, and Green. This happens to be from Chapter 15, in the part on the Edgeworth box:

"We can now verify a simple but important fact: Any Walrasian equilibrium allocation ... necessarily belongs to the Pareto set... Thus, at any competitive allocation ..., there is no alternative feasible allocation that can benefit one consumer without hurting the other. The conclusion that Walrasian allocations yield Pareto optimal allocations is an expression of the first fundamental theorem of welfare economics, a result that ... holds with great generality...

The first fundamental welfare theorem provides, for competitive market economies, a formal expression of Adam Smith's 'invisible hand.' Under perfectly competitive conditions, any equilibrium allocation is a Pareto optimum, and the only possible welfare justification for intervention in the economy is the fulfillment of distributional objectives."

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory, Oxford University Press (1995): p. 524

Adam Smith was little interested in the allocation of given resources, as compared to economic growth. But put aside the dubious historical claim. I want to focus on other reasons why the above passage is nonsense.

What is an "intervention in the economy"? Here are some examples of what could be interventions:

• Refusing to permit contracts in which people sell themselves into slavery.
• Banning child labor.
• Banning the enforcement of clauses in deeds which prohibit owners, in perpetuity, from selling their property to jews, negroes, or members of some other groups.
• Requiring sellers of food in grocery stores to list the amount of the Recommended Daily Allowance of various vitamins and minerals provided by that food.
• Inventing a legal structure in which people can form corporations which limit their legal liability.
• Banning people from copying some books, computer source code, DVDs, and making them freely available or selling them.

Whether to count these as interventions could be viewed as a political question. I would like to think the first four are not current questions. Law provides a background, often taken as given, on which buying and selling can be based. What contracts will be backed up by government varies with time and place.

Some elements of this background are disputed at the moment, at least by those who can attract the attention of the owners of the means of communication. An alteration or decision on a disputed element could be defined as a matter of government intervention in the economy. But such a definition does not seem to have any place in a mathematical theorem.

What counts as property is a question closely related to what counts as an intervention. It is easy easy to write, "Let ω be a vector of endowments..." But whether or not something is an endowment also varies with time, space, and the legal background. Examples that come to my mind without much thinking include air rights in New York City, a capability of a eight-year old to supply so many hours of labor, and wombs in a society where one can contract for surrogate motherhood.

Notice that conventions on contracts and property law shape the distribution of income. The distribution of income is a subject that mainstream economists have been notoriously poorly-trained to discuss.

Mainstream economists may think they are getting a rigorous introduction to economics, what with the "maze of pretentious and unhelpful symbols" (Keynes) in their books. They are also getting, however, a replication of a confused naturalization and reification of the economy common in popular discourse.

Update (27 July 2018): Fixed deleted part of quotation that made nonsense of my point.

An Opportunity For The Working Class With Increased Markups

A Switch-Point Perturbation Diagram

I have a new working paper at SSRN.

Abstract: This article presents an analysis based on a comparison of stationary states. With technology and relative markups among industries taken as exogenous, the long-period trade-off between wages and rates of profits is determined. A long-period change in relative markups among industries can create a switch point exhibiting capital-reversing. Around such a switch point, a higher wage is associated with firms wanting to employ more labor for a given net output – a favorable occurrence for organized labor.

Neoliberals, Liberals, Progressives, Social Democrats, and Democratic Socialists

I guess this is a post on current events in the United States. Some articles I have recently found interesting are:

• Sean Wilentz defends liberals from those further left.
• Gilad Edelman reacts in the Washington Monthly.
• Nathan Robinson on the range of ideas connected with socialism.

I take it socialists want to work towards a society in which, "The free development of each is the condition for the free development of all" (Karl Marx). The idea is that each person should be able to develop their talents to the fullest extent possible, both for their own sake and to contribute as much as possible to society. This is a Christian idea as well, put forth in the parable of the talents (Matthew 25, verses 14-30).

I think traditional conservatives do not agree. They think the vast majority must be consigned to nasty and grubby grunt work. Only those at the top can flourish. Neoliberals put forth a vision of personal development which I find narrow and stilted. In neoliberalism, everybody is an investor developing their human capital for validation by the market. If you have a skill that does not pay - too bad. You wasted your time. For more on this, see Wendy Brown's Undoing the Demos: Neoliberalism's Stealth Revolution.

Somewhere in here I should probably say something about meritocracy. Also, if one wants to look for authoritative Marxist-Leninist accounts of the distinction between socialism and communism, one might look at Marx's private letter, Remarks on the Gotha Program, or Lenin's State and Revolution. But these documents are not what this post is about, since they do not seem relevant to why debates on the post topic are current events.

A crucial question, I think, is whether capitalism could ever be a society in which the free development of all is possible. Social democrats say, "Yes". They think the cruelties of capitalism can be tamed with a generous enough welfare state. One might say, they want a capitalism with a human face. Democratic socialists think not. They want to eventually move beyond capitalism.

I do not see that social democrats and democratic socialists need disagree on immediate, short term programs. Such a tactical coalition can include progressive and liberals. I was curious that none of those three articles linked at the top mentioned Eduard Bernstein. Sure, Otto Von Bismarck created many elements of the first welfare state, as a reactionary response to growing worker power. One might also mention Pope Leo XIII's Rerum Novarum, on the rights and duties of labor. But Bernstein's Marxist reformism - "The movement is everything, the final goal is nothing" - is also important in considering the historical origins of social democracy and democratic socialism.

I was annoyed with Wilentz's suggestion that those further left than him have forgotten John Maynard Keynes. Keynes was historically and globally important in designing the post war Bretton Woods system, a system that brought general prosperity for three decades. For national and international policies, Gunnar Myrdal, Michel Kalecki, Nicholas Kaldor, and Joan Robinson had some influence. Myrdal and Kalecki came to their Keynesianism independently of Keynes.

Where democratic socialists want to go when transcending capitalism is not exactly clear. I do not see that they need to agree. Developing sovereign wealth funds; developing Universal Basic Income (UBI) programs; and supporting labor unions, Employee Stock Ownership Plans (ESOPs), and workers cooperatives are elements of a program fairly radical for these barbaric times.

Update (14 July 2018): A Nicholas Colin article in Medium. There's probably a lot more on-topic current articles.