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Curtis McMullen
Curtis McMullen
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The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would everyever show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving $x^n = a$x^n =a was considered "natural and explicit" because one can find the roots using logarithms. (Set $x = \exp((1/n) \log a$x = exp((1/n) log a).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of $+$, $-$+/-, $*$*, division, and x $x \mapsto x^{1/n}$-> x^(1/n). Obviously Obviously something new has to be added, and and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation $x^n = a$x^n = a is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for $a > 0$a>0, any real initial guess $x > 0$x>0 will converge and the number of digits of accuracy will double with each iteration. For complex $a$a, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero -, indeed its Hausdorff dimension is less than two< 2).

The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would every show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving $x^n = a$ was considered "natural and explicit" because one can find the roots using logarithms. (Set $x = \exp((1/n) \log a$).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of $+$, $-$, $*$, division, and $x \mapsto x^{1/n}$. Obviously something new has to be added, and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation $x^n = a$ is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for $a > 0$, any real initial guess $x > 0$ will converge and the number of digits of accuracy will double with each iteration. For complex $a$, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero - indeed its Hausdorff dimension is less than two).

The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would ever show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving x^n =a was considered "natural and explicit" because one can find the roots using logarithms. (Set x = exp((1/n) log a).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of +/-, *, division, and x -> x^(1/n). Obviously something new has to be added, and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation x^n = a is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for a>0, any real initial guess x>0 will converge and the number of digits of accuracy will double with each iteration. For complex a, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero, indeed Hausdorff dimension < 2).

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Sam Nead
Sam Nead
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The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would every show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving x^n =a$x^n = a$ was considered "natural and explicit" because one can find the roots using logarithms. (Set x = exp((1/n) log a$x = \exp((1/n) \log a$).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of +/-$+$, *$-$, $*$, division, and x -> x^(1/n)$x \mapsto x^{1/n}$. Obviously Obviously something new has to be added, and and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation x^n = a$x^n = a$ is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for a>0$a > 0$, any real initial guess x>0$x > 0$ will converge and the number of digits of accuracy will double with each iteration. For complex a$a$, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero, - indeed its Hausdorff dimension < 2is less than two).

The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would every show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving x^n =a was considered "natural and explicit" because one can find the roots using logarithms. (Set x = exp((1/n) log a).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of +/-, *, division, and x -> x^(1/n). Obviously something new has to be added, and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation x^n = a is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for a>0, any real initial guess x>0 will converge and the number of digits of accuracy will double with each iteration. For complex a, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero, indeed Hausdorff dimension < 2).

The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would every show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving $x^n = a$ was considered "natural and explicit" because one can find the roots using logarithms. (Set $x = \exp((1/n) \log a$).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of $+$, $-$, $*$, division, and $x \mapsto x^{1/n}$. Obviously something new has to be added, and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation $x^n = a$ is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for $a > 0$, any real initial guess $x > 0$ will converge and the number of digits of accuracy will double with each iteration. For complex $a$, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero - indeed its Hausdorff dimension is less than two).

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Curtis McMullen
Curtis McMullen
  • 959
  • 9
  • 12

The interest of algebraic solutions by radicals is a little similar to the interest of geometric solutions by ruler and compass. Part of the charm is that, while it is straightforward (in many cases) to see what can be done, it is mysterious how one would every show something can't be done -- whether that something is squaring the circle, or solving a quintic polynomial.

As for radicals, it is clear from Klein's work that solving x^n =a was considered "natural and explicit" because one can find the roots using logarithms. (Set x = exp((1/n) log a).
The use of transcendental functions here was not at all off-limits, in fact it was one of the main ideas. A "formula" for the roots of a polynomial involving radicals is therefore regarded as an "explicit" (if multivalued) function built using the elementary operations of +/-, *, division, and x -> x^(1/n). Obviously something new has to be added, and the radical operation is one of the simplest after the arithmetic operations.

In his lectures on the icosahedron, Klein similarly has no reservations about using quite sophisticated modular functions just to compute the inverse of a specific degree 60 rational map, which can in turn to be used to solve quintic equations.

In modern terms, one could argue that solving the equation x^n = a is distinguished and routine since Newton's method is highly reliable and rapid in this case. Indeed for a>0, any real initial guess x>0 will converge and the number of digits of accuracy will double with each iteration. For complex a, any initial guess chosen at random will also converge (i.e. the Julia set has measure zero, indeed Hausdorff dimension < 2).