standardized form for products of surds

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As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be for the program to have the property that any two inputs that are equal have the same output. I recognize that I'm unlikely to manage this, but I can get "closer" by placing the tightest restrictions I can on the form the output can take. For example,



  1. No irrational denominators are allowed.

  2. All sums are linearly independent over the rationals.

  3. No products in which one or more of the factors is a sum are allowed.

The question is what kind of restriction I might place on products of surds. For real-valued surds $sqrt[n_1]a_1$,...,$sqrt[n_m]a_m$, a simple way to standardize the form of their product is to consolidate it into a single surd:
$$Pi_j=1^msqrt[n_j]a_j=sqrt[r]Pi_j=1^ma_j^fracrn_j$$
where $r=mathrmLCM(n_1,...,n_m)$. The restriction would then be that



  1. No products in which more than one of the factors is a surd are allowed.

It ensures that, for example, the inputs $sqrt3sqrt[3]2$ and $sqrt[6]3sqrt[3]6$ yield the common output $sqrt[6]108$. Together with restriction 3, it implies that the only allowed products are ones between one rational number and one surd.



However, since the program allows arbitrary radical expressions as inputs, I have to deal with complex-valued surds** too, and for them, the identity given above doesn't hold - I believe it's true up to multiplication by an appropriate root of unity. I tried working out a concrete example to see what it would look like:
$$sqrt-3sqrt[3]2=e^fracpi i3sqrt[6]-27*4=(frac12+fracsqrt-32)sqrt[6]-108$$
Unfortunately, given restriction 3, I would then have to resolve $fracsqrt-32sqrt[6]-108$, and the raising of the index of $sqrt-3$ from $2$ to $6$ necessary to consolidate this product would require another multiplication by $e^fracpi i3$, so that the sequence of manipulations would never terminate.



So first of all, is it in general impossible to enforce both restriction 3 and restriction 4 when complex-valued surds are involved, or does doing so just require being craftier than I was above? If the former, what is the best I could hope to do in enforcing a standard form for products of surds?



*I use the term "radical expression" to refer to the smallest set $S$ of complex numbers such that



  • $mathbbQsubset S$

  • If $a,bin S$, then $a+b,a-b,a*bin S$

  • If $a,bin S$ and $bne0$, then $fracabin S$

  • If $ain S$, $binmathbbQ$, and $a$ and $b$ are not both $0$, then $a^bin S$

The term also implies expression purely in terms of a finite combination of integers, arithmetic operations, and root extractions.



**I use the convention where $sqrt[n]a$ refers to the root of $a$ whose argument is $fracmathrmArg(a)n$ and $mathrmArg(a)$ is defined to be the argument of $a$ that lies in $[0,2pi)$.




complex-numbers soft-question radicals




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    As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be for the program to have the property that any two inputs that are equal have the same output. I recognize that I'm unlikely to manage this, but I can get "closer" by placing the tightest restrictions I can on the form the output can take. For example,



    1. No irrational denominators are allowed.

    2. All sums are linearly independent over the rationals.

    3. No products in which one or more of the factors is a sum are allowed.

    The question is what kind of restriction I might place on products of surds. For real-valued surds $sqrt[n_1]a_1$,...,$sqrt[n_m]a_m$, a simple way to standardize the form of their product is to consolidate it into a single surd:
    $$Pi_j=1^msqrt[n_j]a_j=sqrt[r]Pi_j=1^ma_j^fracrn_j$$
    where $r=mathrmLCM(n_1,...,n_m)$. The restriction would then be that



    1. No products in which more than one of the factors is a surd are allowed.

    It ensures that, for example, the inputs $sqrt3sqrt[3]2$ and $sqrt[6]3sqrt[3]6$ yield the common output $sqrt[6]108$. Together with restriction 3, it implies that the only allowed products are ones between one rational number and one surd.



    However, since the program allows arbitrary radical expressions as inputs, I have to deal with complex-valued surds** too, and for them, the identity given above doesn't hold - I believe it's true up to multiplication by an appropriate root of unity. I tried working out a concrete example to see what it would look like:
    $$sqrt-3sqrt[3]2=e^fracpi i3sqrt[6]-27*4=(frac12+fracsqrt-32)sqrt[6]-108$$
    Unfortunately, given restriction 3, I would then have to resolve $fracsqrt-32sqrt[6]-108$, and the raising of the index of $sqrt-3$ from $2$ to $6$ necessary to consolidate this product would require another multiplication by $e^fracpi i3$, so that the sequence of manipulations would never terminate.



    So first of all, is it in general impossible to enforce both restriction 3 and restriction 4 when complex-valued surds are involved, or does doing so just require being craftier than I was above? If the former, what is the best I could hope to do in enforcing a standard form for products of surds?



    *I use the term "radical expression" to refer to the smallest set $S$ of complex numbers such that



    • $mathbbQsubset S$

    • If $a,bin S$, then $a+b,a-b,a*bin S$

    • If $a,bin S$ and $bne0$, then $fracabin S$

    • If $ain S$, $binmathbbQ$, and $a$ and $b$ are not both $0$, then $a^bin S$

    The term also implies expression purely in terms of a finite combination of integers, arithmetic operations, and root extractions.



    **I use the convention where $sqrt[n]a$ refers to the root of $a$ whose argument is $fracmathrmArg(a)n$ and $mathrmArg(a)$ is defined to be the argument of $a$ that lies in $[0,2pi)$.




    complex-numbers soft-question radicals




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      As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be for the program to have the property that any two inputs that are equal have the same output. I recognize that I'm unlikely to manage this, but I can get "closer" by placing the tightest restrictions I can on the form the output can take. For example,



      1. No irrational denominators are allowed.

      2. All sums are linearly independent over the rationals.

      3. No products in which one or more of the factors is a sum are allowed.

      The question is what kind of restriction I might place on products of surds. For real-valued surds $sqrt[n_1]a_1$,...,$sqrt[n_m]a_m$, a simple way to standardize the form of their product is to consolidate it into a single surd:
      $$Pi_j=1^msqrt[n_j]a_j=sqrt[r]Pi_j=1^ma_j^fracrn_j$$
      where $r=mathrmLCM(n_1,...,n_m)$. The restriction would then be that



      1. No products in which more than one of the factors is a surd are allowed.

      It ensures that, for example, the inputs $sqrt3sqrt[3]2$ and $sqrt[6]3sqrt[3]6$ yield the common output $sqrt[6]108$. Together with restriction 3, it implies that the only allowed products are ones between one rational number and one surd.



      However, since the program allows arbitrary radical expressions as inputs, I have to deal with complex-valued surds** too, and for them, the identity given above doesn't hold - I believe it's true up to multiplication by an appropriate root of unity. I tried working out a concrete example to see what it would look like:
      $$sqrt-3sqrt[3]2=e^fracpi i3sqrt[6]-27*4=(frac12+fracsqrt-32)sqrt[6]-108$$
      Unfortunately, given restriction 3, I would then have to resolve $fracsqrt-32sqrt[6]-108$, and the raising of the index of $sqrt-3$ from $2$ to $6$ necessary to consolidate this product would require another multiplication by $e^fracpi i3$, so that the sequence of manipulations would never terminate.



      So first of all, is it in general impossible to enforce both restriction 3 and restriction 4 when complex-valued surds are involved, or does doing so just require being craftier than I was above? If the former, what is the best I could hope to do in enforcing a standard form for products of surds?



      *I use the term "radical expression" to refer to the smallest set $S$ of complex numbers such that



      • $mathbbQsubset S$

      • If $a,bin S$, then $a+b,a-b,a*bin S$

      • If $a,bin S$ and $bne0$, then $fracabin S$

      • If $ain S$, $binmathbbQ$, and $a$ and $b$ are not both $0$, then $a^bin S$

      The term also implies expression purely in terms of a finite combination of integers, arithmetic operations, and root extractions.



      **I use the convention where $sqrt[n]a$ refers to the root of $a$ whose argument is $fracmathrmArg(a)n$ and $mathrmArg(a)$ is defined to be the argument of $a$ that lies in $[0,2pi)$.




      complex-numbers soft-question radicals




      share|cite|improve this question











      As an exercise, I'm writing a program that takes a radical expression* as an input and returns a radical expression equal to it that is in as close to a canonical form as possible. The ideal would be for the program to have the property that any two inputs that are equal have the same output. I recognize that I'm unlikely to manage this, but I can get "closer" by placing the tightest restrictions I can on the form the output can take. For example,



      1. No irrational denominators are allowed.

      2. All sums are linearly independent over the rationals.

      3. No products in which one or more of the factors is a sum are allowed.

      The question is what kind of restriction I might place on products of surds. For real-valued surds $sqrt[n_1]a_1$,...,$sqrt[n_m]a_m$, a simple way to standardize the form of their product is to consolidate it into a single surd:
      $$Pi_j=1^msqrt[n_j]a_j=sqrt[r]Pi_j=1^ma_j^fracrn_j$$
      where $r=mathrmLCM(n_1,...,n_m)$. The restriction would then be that



      1. No products in which more than one of the factors is a surd are allowed.

      It ensures that, for example, the inputs $sqrt3sqrt[3]2$ and $sqrt[6]3sqrt[3]6$ yield the common output $sqrt[6]108$. Together with restriction 3, it implies that the only allowed products are ones between one rational number and one surd.



      However, since the program allows arbitrary radical expressions as inputs, I have to deal with complex-valued surds** too, and for them, the identity given above doesn't hold - I believe it's true up to multiplication by an appropriate root of unity. I tried working out a concrete example to see what it would look like:
      $$sqrt-3sqrt[3]2=e^fracpi i3sqrt[6]-27*4=(frac12+fracsqrt-32)sqrt[6]-108$$
      Unfortunately, given restriction 3, I would then have to resolve $fracsqrt-32sqrt[6]-108$, and the raising of the index of $sqrt-3$ from $2$ to $6$ necessary to consolidate this product would require another multiplication by $e^fracpi i3$, so that the sequence of manipulations would never terminate.



      So first of all, is it in general impossible to enforce both restriction 3 and restriction 4 when complex-valued surds are involved, or does doing so just require being craftier than I was above? If the former, what is the best I could hope to do in enforcing a standard form for products of surds?



      *I use the term "radical expression" to refer to the smallest set $S$ of complex numbers such that



      • $mathbbQsubset S$

      • If $a,bin S$, then $a+b,a-b,a*bin S$

      • If $a,bin S$ and $bne0$, then $fracabin S$

      • If $ain S$, $binmathbbQ$, and $a$ and $b$ are not both $0$, then $a^bin S$

      The term also implies expression purely in terms of a finite combination of integers, arithmetic operations, and root extractions.



      **I use the convention where $sqrt[n]a$ refers to the root of $a$ whose argument is $fracmathrmArg(a)n$ and $mathrmArg(a)$ is defined to be the argument of $a$ that lies in $[0,2pi)$.





      complex-numbers soft-question radicals





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