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The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of the Euclidean domain. It isn't obvious to me how this is the case. Could you provide a palatable explanation?

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Euclidean domains are rings that can be endowed with the structure of an Euclidean function. An example is the ring of polynomials $\mathbb{K}(X)$ over a field $\mathbb{K}$, endowed with the Euclidean function $f(P)=\mathrm{deg}P$, $P\in\mathbb{K}(X)$. –  Sanath Devalapurkar Apr 24 '14 at 1:49
I've written on this in section 1 and 4 here: onlinemathcircle.com/wp-content/uploads/2011/03/13-polynum.pdf –  Holden Lee Apr 24 '14 at 21:31

3 Answers 3

up vote 5 down vote accepted

absolute value of integer <-> degree of polynomial

positive integer <-> monic polynomial

+/- 1 <-> constant polynomial

prime integer <-> irreducible polynomial

With these correspondences, there are many identical notions and theorems, like the division algorithm, unique prime factorization, principal ideals, LCM, GCD, ...

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The Euclidean algorithm has already been mentioned. Related to this, one has:

  • Non-zero integers admit unique factorization into primes (up to sign).

  • Non-zero polynomials admit unique factorization into irreducibles (up to non-zero constants).

  • The integers modulo the ideal generated by any prime form a field.

  • Polynomials over a field modulo the ideal generated by any irreducible polynomial form a field.

  • Any ideal in either of the rings is principal.
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Euclid's algorithm can find the gcd of two integers. A natural variation on Euclid's classic algorithm can find the gcd of two polynomials. Among similarities, that's the big one.

Integers, and polynomials, are closed under addition, subtraction, and multiplication, but not under division.

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