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It is a question about abstract algebra especially about rings.

In the book named 'A first course in abstract algebra by John B. Fraleigh',

the definition of Polynomials is infinite formal sum.

because defining Polynomials by finite formal sum cause a trouble.

In finite formal sum definition '0 + ax' and '0 + ax + 0x^2' are different as formal sums.

but we want to regard them as the same polynomial.

I think Fraleigh's book's explanation is reasonable but in some textbooks (dummit, pinter)

definitions of polynomials are formal finite sum.

why are some texts just defining polynomials like that?

Is there any no trouble in the end?

thank you for reading this question.

I used translater.

ju so
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    You can use a finite sum definition if you add the condition that the highest degree term has a non-zero coefficient. That makes the polynomial representation unique and avoids the problem. With the infinite sum definition you need to add the condition that only a finite number of coefficients are non-zero. They are both valid definitions. You can use whichever you like best. – Jaap Scherphuis Oct 11 '22 at 08:34
  • @JaapScherphuis thank you!. I understand what you said. but when using finite formal sum, many texts define addition of two polynomials like this. if one polynomial degree is less than the other one, then produce terms in which coefficients are zero until the degree is the same. then add two polynomial pointwise. so this procedure of addition let polynomial's highest degree term has a zero coefficient. what do you think? – ju so Oct 11 '22 at 09:26
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    That's a good point. When adding polynomials you want to temporarily allow "improper polynomials" with extra zero-coefficient terms to make the process work. In a way this is similar to the improper fractions you get when adding fractions - $\frac14+\frac14=\frac24=\frac12$. Here $\frac24$ and $\frac12$ mean the same but one is the canonical representation. Using the infinite sum definition of polynomial makes the addition process more straightforward. The extra zero terms are always available instead of included only when convenient. Of course we still only write them when needed. – Jaap Scherphuis Oct 11 '22 at 09:45
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    There are many possible representations of (formal) polynomials. But what is important is not this particular syntax but rather the associated semantics (meaning), i.e. they have only finitely many nonzero coefficients (excludes formal power series), and two polynomials are equal iff all their corresponding coef's are equal. For more see here and here. – Bill Dubuque Oct 11 '22 at 12:42
  • @JaapScherphuis thanks. I think i understand almost. but still not 100% clear. I heard that we can think 2/4, 1/2 are in the same equivalence class. then Is it possible making equivalence class in which '0 + ax' and '0 + ax + 0x^2' are elements?. and seperately, do you know what 'formal sum' means? what 'formal' means? it means 'expression'? – ju so Oct 11 '22 at 15:46

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