Einstein’s Theory of Irreducible Algebraic Polynomials
Uploaded by jamie83 on Oct 26, 2011
This paper discusses Einstein’s theory of irreducible polynomials in algebra. (6+ pages; 4 sources; MLA citation style)
I Introduction
Mathematicians like Einstein seek to explain how the world works; their tools for doing so are the laws of mathematics.
Einstein is probably best known for his work on relativity, and the Unified Field Theory, but he did significant work in other areas of mathematics as well.
This paper will discuss his theory with regard to polynomials used in algebra, and why they are irreducible. As a non-mathematician, the only way I can hope to approach this is to reproduce the theory itself, and then define the terms used to formulate it. By restating the terms in my own words, I can then work toward a better understanding of the theory.
II Einstein’s Irreducible Criterion
This is the criterion Einstein demanded for irreducible polynomials in algebra:
“A sufficient condition assuring that an integer polynomial p(x) is irreducible in the polynomial ring .
The polynomial
where for all and (which means that the degree of p(x) is n) is irreducible if some prime number p divides all coefficients , but not the leading coefficient and, moreover, does not divide the constant term .
This is only a sufficient, and by no means a necessary condition. For example, the polynomial is irreducible, but does not fulfil the above property, since no prime number divides 1.” (Barile, PG).
To me, this is hopeless! But perhaps defining the terms will help. We need to understand what is meant by “sufficient condition”, “necessary condition,” “integer polynomial”, “irreducible”, and “polynomial ring”. But before that, let’s look at what polynomials are.
III Polynomials
Polynomials are mathematical expressions of the type “3x2 +2x +2”; a series of “terms” that describe a condition we wish to solve; they are basically sums of other expressions. The “3x2” is referred to as the “leading term” and the “2” is the constant term, because it has no exponent or any other symbol indicating modification; 2 is always 2. (Stapel, PG). Polynomials are arranged according to their exponents, with the highest first: the expression “2x + 3 + 7x2” would be rearranged to be written “7x2 + 2x + 3”. And because the first term’s exponent is a square, this is a second-degree polynomial. If we had the expression “7x5 + 2x +3,” we would...