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