This paper discusses Einstein’s theory of irreducible polynomials in algebra. (6+ pages; 4 sources; MLA citation style)
IIntroduction
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.
IIEinstein’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 .
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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.
IIIPolynomials
Polynomials are mathematical expressions of the type “3×2 +2x +2”; a series of “terms” that describe a condition we wish to solve; they are basically sums of other expressions. The “3×2” 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 + 7×2” would be rearranged to be written “7×2 + 2x + 3”. And because the first term’s exponent is a square, this is a second-degree polynomial. If we had the expression “7×5 + 2x +3,” we would have a fifth degree polynomial.
In algebra, it’s desirable to “reduce” polynomial expressions to their simplest form, by combining like terms; basically we think of it as solving the equation. If we had something like “2×2 +3x +1 – x2 + 4×2 –x” we wouldn’t leave it like that; we’d reduce it to a much simpler form: we’d put the like terms together, and “solve” it, like this: “(2×2 –x2 +4×2) + (3x –x) +1” = “5×2 + 2x +1.” At this point, the equation can’t be reduced any further (x2 and x are not the same term; the “x” won’t go away); it is irreducible. (I suppose we could divide through by x, but that would leave something like 5x +2 +1/x, which doesn’t seem any improvement.) At any rate, my shaky math aside, “irreducible” means that it’s not possible to factor the equation down any further; it is in its simplest form.
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IVBack to Einstein
Now we know what polynomials are, and worked a bit with irreducible polynomials. How does that relate to Einstein’s theorem, which I’ve given above?
First, Einstein says that there must be a “sufficient condition” for his theory to hold true. That means we have to define “condition” as well, then modify it. A “condition” is a requirement that has to be in place before a theorem can be held to be true. A “necessary” condition is one that must hold for something to be true but doesn’t guarantee that it’s true. A “sufficient” condition is one that guarantees that if it’s true, the result is also true. It is therefore a stronger condition than a “necessary” one. (I mention them both because they’re often used together: a “necessary and sufficient” condition, but they’re not identical.) (Weisstein, PG).
Here’s a statement about conditions: “…the condition that a decimal number n end in the digit 2 is a sufficient but not necessary condition that n be even.” In other words, if a number ends in 2 (2, 12, 22, 32, 372, whatever), that is a sufficient condition to tell us that the number is even. We are postulating the idea that a number, any number, ending in 2 is even, and if that is sufficient, as it is, then we can state that any such number is even.
We need now to define an “integer polynomial” and a “polynomial ring.” An “integer polynomial” is a polynomial wherein all the coefficients are integers. (In the example anx, an is the coefficient; if n = 2, this term become 2x.)
A “polynomial ring” is a very complex set of numbers coupled with specific operations. In general, the mathematical computations done in a ring are circular, leading back to the beginning, hence the name. In formal terms, a ring is a set “S” of numbers, which, when considered together with two binary operators, usually addition and multiplication (+ and *), satisfies the following conditions: It has “additive associativity” such that for all a, b, and c belonging to set S, (a + b) + c = a + (b + c).
Example: (2 + 3) + 4 = 2 + (3 + 4).
No matter which way we work the problem, the answer is the same:
(2 + 3) + 4 = 5 + 4 = 9; or 2 + (3 + 4) = 2 + 7 = 9.
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It has “additive commutativity” such that for all a and b in set S, a + b = b + a.
It has “additive identity” so that there exists “an element 0” within set S, such that for all a within S, 0 + a = a + 0 = a. (Zero added to anything results in the same number.) (“Ring,” PG).
It has “additive inverse,” so that for every a within set S there exists a negative (-a), such that a + (-a) = (-a) + a = 0.
It has “multiplicative associativity,” such that for all a, b and c within S, (a*b)*c = a*(b*c).
That is, (1*2)*3 = 1*(2*3).
Solving: (1*2)*3 = (2)*3 = 6; or 1(2*3) = 1(6) = 6.
Finally, the ring has “left and right distributivity, such that for all a, b and c within S, a*(b+c) = (a*b) + (a*c) and (b+c)*a = (b*a) + (c*a).
This one works as well. Let a = 2, b =3 and c = 4. Then 2*(3 + 4) = (2*3) + (2*4) = 2*(7) = 6 + 8 = 14 = 14; the opposite is correct as well. (“Ring,” PG).
Now, can we go back and look at Einstein’s example? Recall, his theory looks for “a sufficient condition assuring that an integer polynomial is irreducible in the polynomial ring.” He says that the polynomial p(x) (below) meets the condition:
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 .
In other words, the integer “ai” is a number within the set Z, wherein all i (integers) are between 0 and n (all positive); further, at no time is the coefficient an equal to 0; and finally, if some prime number (a number that itself cannot be further divided) divides all the coefficients except the leading one; and if that same prime number squared does not divide the constant; the polynomial will satisfy the condition. From here, it would seem the best course of action would be to “plug” in some numbers and test the equation against the six properties of a ring.
IVConclusion
Unfortunately, it is beyond me to solve this equation and prove Einstein’s theorem. But we can see that he’s done a couple of things that make it easier: first, he works only with positive numbers, thus eliminating the hassles that come with negative numbers. Secondly, he works only with integers (whole numbers); and finally, he uses prime numbers to divide through (though he doesn’t use 1).
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In every case, he begins from the simplest step, and continues to reduce the equations until finally they are in fact irreducible.
VReferences
Barile, Margherita. “Einstein’s Irreducibility Criterion.” Mathworld. [Website]. Dated: 1999-2003. http://mathworld.wolfram.com/EisensteinsIrreducibilityCriterion.html
Stapel, Elizabeth. “Polynomials.” Purplemath [Website]. Dated: 2000-2003. Accessed: 4 Oct 2003. http://www.purplemath.com/modules/polydefs.htm
Weisstein, Eric. “Condition.” Mathworld [Website]. 1999-2003. Accessed: 4 Oct 2003. http://mathworld.wolfram.com/Condition.html
“Ring.” Mathworld [Website]. 1999-2003. Accessed: 4 Oct 2003. http://mathworld.wolfram.com/Ring.html
