# Application of the fundamental theorem of algebra

I'm just going through my lecture notes and under one subheading "application", I have written:

If $m = \prod_{i = 1}^k p_i^{r_i}$ and $n = \prod_{i = 1}^k p_i^{s_i}$, for primes $p_i$ & $r_i, s_i \in \mathbb{N}$. Then $\gcd(m,n) = \prod_{i = 1}^k p_i^{\min(r_i,s_i)}$ and $\mathrm{lcm}(m,n) = \prod_{i = 1}^k p_i^{\min(r_i,s_i)}$.

Should they both be $\min(r_i,s_i)$, because this just gives gcd = lcm doesn't it?

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The second is a typo for max – André Nicolas Jan 30 '13 at 20:32
Do you mean the fundamental theorem of arithmetic? – Jonas Meyer Jan 30 '13 at 20:34
@JonasMeyer I swear my lecturer said that but then she (definitely) wrote "algebra" on the board, so I just thought maybe it has two names? – Kaish Jan 30 '13 at 20:37
Just googled Fundamental Theorem of Algebra and it is nothing like theory of arithmetic lol. – Kaish Jan 30 '13 at 20:38
@Kaish: Please: TeX is sophisticated. Write \prod_{i=1}^k, not \Pi_{i=1}^k. These don't look the same $\displaystyle\prod_{i=1}^k$ versus $\displaystyle\Pi_{i=1}^k$. Also, don't write \mathrm{min}; just write \min. Look: $\displaystyle\min_{a\in A} f(a)$ versus $\displaystyle\mathrm{min}_{a\in A} f(a)$. The latter form also fails to provide proper spacing before and after, as in $a\min b$. And \gcd is also a standard operator name in LaTeX. – Michael Hardy Jan 30 '13 at 21:24

The second one is a typo, it should read $\max$.
The Fundamental Theorem of Algebra has various statements. One of them is that a non-constant polynomial with coefficients in $\mathbb{C}$, the set of complex numbers, always has a root in $\mathbb{C}$.
It can be viewed as a factorization theorem, because it is equivalent to the result that a polynomial of degree $\ge 1$ over $\mathbb{C}$ can be expressed as a product of linear polynomials with coefficients in $\mathbb{C}$.