text: # Order of point in divisor
  
  Let $y^2=f(x)$ be hyperelliptic curve over $k$ and $(a,\sqrt{f(a)})$ point on the curve and let $b\in k$. I would like to prove that divisor of the function $g(x,y)=\frac{x-b}{x-a}$ is equal to
  
  $$div(g)=(b,\sqrt{f(b)})+(b,-\sqrt{f(b)})-(a,\sqrt{f(a)})-(a,-\sqrt{f(a)}).$$
  
  It is easy to show that first two points are the only zero points and other two are the only poles, we also have point in infinity $(0:1:0)$ but $g$ has value not equal to $0$ in that point so order of point of infinity is $0$ and is not in divisor.
  
  I can also show that, for example, the uniformizer in $P=(b,\sqrt{f(b)})$ is $y-\sqrt{f(b)}$ bu I don't know how to use it to show that order of that point is $1$. I also know that $ord_P(g)=ord_P(x-b)+ord_p(x-a)=ord_P(x-b)$ since $x-a$ doesn't vanish in $P$. If my curve was $\mathbb{P}^2$ I know I could conclude it is $1$ since it is zero point with multiplicity $1$? But what can I conclude here?
  
  EDIT: I forgot to write that $(a,\sqrt{f(a)})$ is the point on curve in field $k$ and that could maybe help for some ideas?
explanation: The text extract is a detailed mathematical discussion about divisors on a hyperelliptic curve. It presents a specific problem and provides some steps towards a solution, but it does not contain a clear, direct question or a complete answer. Instead, it outlines a problem and some partial reasoning, leaving the reader to infer the remaining steps or questions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Order of point in divisor
  
  Let $y^2=f(x)$ be hyperelliptic curve over $k$ and $(a,\sqrt{f(a)})$ point on the curve and let $b\in k$. I would like to prove that divisor of the function $g(x,y)=\frac{x-b}{x-a}$ is equal to
  
  $$div(g)=(b,\sqrt{f(b)})+(b,-\sqrt{f(b)})-(a,\sqrt{f(a)})-(a,-\sqrt{f(a)}).$$
  
  It is easy to show that first two points are the only zero points and other two are the only poles, we also have point in infinity $(0:1:0)$ but $g$ has value not equal to $0$ in that point so order of point of infinity is $0$ and is not in divisor.
  
  I can also show that, for example, the uniformizer in $P=(b,\sqrt{f(b)})$ is $y-\sqrt{f(b)}$ bu I don't know how to use it to show that order of that point is $1$. I also know that $ord_P(g)=ord_P(x-b)+ord_p(x-a)=ord_P(x-b)$ since $x-a$ doesn't vanish in $P$. If my curve was $\mathbb{P}^2$ I know I could conclude it is $1$ since it is zero point with multiplicity $1$? But what can I conclude here?
  
  EDIT: I forgot to write that $(a,\sqrt{f(a)})$ is the point on curve in field $k$ and that could maybe help for some ideas?
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a detailed mathematical discussion about divisors on a hyperelliptic curve. It presents a specific problem and provides some steps towards a solution, but it does not contain a clear, direct question or a complete answer. Instead, it outlines a problem and some partial reasoning, leaving the reader to infer the remaining steps or questions.
  
  Label: no-question