stylistic:style_transfer
Definition and examples

The polynomials of a single variable x {\displaystyle x}, with integer coefficients, form a free abelian group under polynomial addition, with the powers of x {\displaystyle x} as a basis. As an abstract group, this is the same as (an isomorphic group to) the multiplicative group of positive rational numbers. One way to map these two groups to each other, showing that they are isomorphic, is to reinterpret the exponent of the i {\displaystyle i} th prime number in the multiplicative group of the rationals as instead giving the coefficient of x i − 1 {\displaystyle x^{i-1}} in the corresponding polynomial, or vice versa. For instance the rational number 5 / 27 {\displaystyle 5/27} has exponents of 0, − 3, 1 {\displaystyle 0,-3,1} for the first three prime numbers 2, 3, 5 {\displaystyle 2,3,5} and would correspond in this way to the polynomial − 3 x + x 2 {\displaystyle -3x+x^{2}} having the same coefficients 0, − 3, 1 {\displaystyle 0,-3,1} for its constant, linear, and quadratic terms. Because these mappings merely reinterpret the same numbers, they define a bijection between the elements of the two groups. And because the group operation of multiplying positive rationals acts additively on the exponents of the prime numbers, in the same way that the group operation of adding polynomials acts on the coefficients of the polynomials, these maps preserve the group structure; they are homomorphisms. A bijective homomorphism is called an isomorphism, and its existence demonstrates that these two groups have the same properties.