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I understand how to show that $x^2+2x+2$ and $x^2+1$ are relatively prime in the field $\mathbb Q$ using the division algorithm, but for the same two polynomials, how do I apply the division algorithm if our field is $\mathbb C$?

Akababa
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Mr.Fry
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    Same calculation, exactly. – André Nicolas Jul 03 '13 at 17:38
  • Cool thanks. Can you just tell me why? Sorry this is a summer course so the details are a but rushed. – Mr.Fry Jul 03 '13 at 17:42
  • Ok, just take a look at your textbook, and see when (i.e, in what kind of ring) we can appply division algorithm to prove, or disprove the co-primity of 2 elements. Do $\mathbb{Z}$, $\mathbb{Q}[x]$, $\mathbb{R}[x]$, and $\mathbb{C}[x]$ satisfy that condition? And why? – user49685 Jul 03 '13 at 17:53
  • The polynomial you are dividing by is monic, so if we start with integer coefficients, all calculations stay in the integers. Or here is some slightly fancier algebra, If two polynomials $A(x)$, $B(x)$ with rational coefficients are relatively prime over $\mathbb{Q}$, there are polynomials $S(x)$, $T(x)$ such that $AS+BT=1$. But then $AS+BT=1$ holds for all complex numbers, so the two polynomials are relatively prime over $\mathbb{C}$ also. – André Nicolas Jul 03 '13 at 17:57

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You approach it in precisely the same manner as you did for the field $\mathbb Q$.

Search your text, and check exactly when (in which sort of ring) we can apply the division algorithm to prove that two elements are coprime (or not).

You'll see that $\;\mathbb Z[x],\, \Bbb Q[x],\,\mathbb R[x],\,$ and $\mathbb C[x]$ each satisfy the condition. Be sure you understand why this is true. The polynomial we're dividing by is monic: when we're working with integer coefficients, all calculations stay in the integers. More generally, any two polynomials which are relatively prime over $\mathbb Q$ are also relatively prime over $\mathbb R$ and $\mathbb C$.

amWhy
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