Find a polynomial with integer coefficients whose solution is the multiplication of the solutions to other...
Let $ax^2 + bx + c = 0$ and $dy^2 + ey + f = 0$ such that variables $a$ through $f$ are fixed integer parameters. I'm trying to find integers $g$, $h$, and $j$ such that $g(xy)^2 + h(xy) + j = 0$. I imagine the equation will be different depending on which roots of the original polynomials we consider. I'm looking for a proof either they exist or don't exist and if they do exist what they are in terms of the original integer values.
The motivation is that I'm trying to represent certain numbers using the form $x = [S, A, B, C]$ where $S$ denotes which root $x$ is to a quadratic polynomial $Ax^2 + Bx + C$. I'm trying to figure out how to multiply such numbers.
algebra-precalculus
|
show 3 more comments
Let $ax^2 + bx + c = 0$ and $dy^2 + ey + f = 0$ such that variables $a$ through $f$ are fixed integer parameters. I'm trying to find integers $g$, $h$, and $j$ such that $g(xy)^2 + h(xy) + j = 0$. I imagine the equation will be different depending on which roots of the original polynomials we consider. I'm looking for a proof either they exist or don't exist and if they do exist what they are in terms of the original integer values.
The motivation is that I'm trying to represent certain numbers using the form $x = [S, A, B, C]$ where $S$ denotes which root $x$ is to a quadratic polynomial $Ax^2 + Bx + C$. I'm trying to figure out how to multiply such numbers.
algebra-precalculus
1
That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13
|
show 3 more comments
Let $ax^2 + bx + c = 0$ and $dy^2 + ey + f = 0$ such that variables $a$ through $f$ are fixed integer parameters. I'm trying to find integers $g$, $h$, and $j$ such that $g(xy)^2 + h(xy) + j = 0$. I imagine the equation will be different depending on which roots of the original polynomials we consider. I'm looking for a proof either they exist or don't exist and if they do exist what they are in terms of the original integer values.
The motivation is that I'm trying to represent certain numbers using the form $x = [S, A, B, C]$ where $S$ denotes which root $x$ is to a quadratic polynomial $Ax^2 + Bx + C$. I'm trying to figure out how to multiply such numbers.
algebra-precalculus
Let $ax^2 + bx + c = 0$ and $dy^2 + ey + f = 0$ such that variables $a$ through $f$ are fixed integer parameters. I'm trying to find integers $g$, $h$, and $j$ such that $g(xy)^2 + h(xy) + j = 0$. I imagine the equation will be different depending on which roots of the original polynomials we consider. I'm looking for a proof either they exist or don't exist and if they do exist what they are in terms of the original integer values.
The motivation is that I'm trying to represent certain numbers using the form $x = [S, A, B, C]$ where $S$ denotes which root $x$ is to a quadratic polynomial $Ax^2 + Bx + C$. I'm trying to figure out how to multiply such numbers.
algebra-precalculus
algebra-precalculus
asked Dec 13 at 5:00
The Great Duck
11732047
11732047
1
That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13
|
show 3 more comments
1
That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13
1
1
That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13
|
show 3 more comments
2 Answers
2
active
oldest
votes
There is a generic polynomial that will get you the products of the roots - but you can't keep the degree the same. After all, there's not necessarily any way to distinguish between the roots, so if one product appears as a root, all possible products of roots do. For two quadratics, that means that we're looking at a fourth-degree polynomial.
For simplicity, I'll work with two monic quadratics, $x^2+ax+b$ and $x^2+cx+d$, with roots $r_1,r_2$ and $s_1,s_2$ respectively. We have $a=-(r_1+r_2)$, $b=r_1r_2$, $c=(-s_1+s_2)$, $d=s_1s_2$, and we want to find the symmetric functions of the products of roots to get the quartic $x^4+ex^3+fx^2+gx+h$.
First, the constant term $h=(r_1s_1)(r_1s_2)(r_2s_1)(r_2s_2)=r_1^2r_2^2s_1^2s_2^2=b^2d^2$.
Second, the $x^3$ term $e=-(r_1s_1+r_1s_2+r_2s_1+r_2s_2)=-(r_1+r_2)(s_1+s_2)=-ac$.
Third, the $x$ term
begin{align*}g &=-r_1s_1r_1s_2r_2s_1-r_1s_1r_1s_2r_2s_2-r_1s_1r_2s_1r_2s_2-r_1s_2r_2s_1r_2s_2\
&=-r_1r_2s_1s_2(r_1s_1+r_1s_2+r_2s_1+r_2s_2)\
&= -abcdend{align*}
Finally, the $x^2$ term
begin{align*}f &= r_1s_1r_1s_2+r_1s_1r_2s_1+r_1s_1r_2s_2+r_1s_2r_2s_1+r_1s_2r_2s_2+r_2s_1r_2s_2\
&= (r_1^2+r_2^2)s_1s_2+r_1r_2(s_1^2+s_2^2)+2r_1r_2s_1s_2\
&= (a^2-2b)d+(c^2-2d)b+2bd = a^2d+c^2b-2bdend{align*}
Putting those together, the final polynomial is
$$x^4 -ac x^3 + (a^2d+c^2b-2bd)x^2 -abcd x + b^2d^2$$
If we tried to do the same with fewer of the products, to build a quadratic, it just wouldn't work. The building blocks we have are the symmetric polynomials in $r_1,r_2$ and $s_1,s_2$, so anything we build with them will have to be symmetric as well; if $r_1s_1$ is a root, we can exchange $r_1$ and $r_2$ to get $r_2s_1$ as another root, or exchange $s_1$ and $s_2$ to get $r_1s_2$ as a third root, or exchange both to get $r_2s_2$ as a fourth root. It's all four or none. We've shown that getting just those four roots is possible, so that's the best that can be done.
In fact, it's typical that for two irreducible quadratics, the quartic we build this way will be irreducible as well. We'd need a coincidence like the discriminants being the same (up to multiplication by a perfect square) to avoid it.
add a comment |
Here's one way to understand why such $g,h,j$ don't necessarily exist:
The roots of an integer quadratic are expressible as
$$frac{xpm sqrt{y}}{z}$$
for some integers $x,y,z$ (say, by the quadratic formula). If you multiply two of these together, you can get something like this:
$$left(frac{x_1+sqrt{y_1}}{z_1}right)left(frac{x_2+sqrt{y_2}}{z_2}right)=frac{x_1x_2+x_1sqrt{y_2}+x_2sqrt{y_1}+sqrt{y_1y_2}}{z_1z_2}.$$
We need these three square-root terms to somehow collapse into one for this to be expressible as $frac{X+sqrt{Y}}{Z}$ for some integers $X,Y,Z$. If there are no nontrivial relations between $sqrt{y_1}$ and $sqrt{y_2}$ (say, if $y_1=2$ and $y_2=3$), then we can't simplify the above at all, and thus it isn't a root of an integer-coefficient quadratic.
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
add a comment |
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2 Answers
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There is a generic polynomial that will get you the products of the roots - but you can't keep the degree the same. After all, there's not necessarily any way to distinguish between the roots, so if one product appears as a root, all possible products of roots do. For two quadratics, that means that we're looking at a fourth-degree polynomial.
For simplicity, I'll work with two monic quadratics, $x^2+ax+b$ and $x^2+cx+d$, with roots $r_1,r_2$ and $s_1,s_2$ respectively. We have $a=-(r_1+r_2)$, $b=r_1r_2$, $c=(-s_1+s_2)$, $d=s_1s_2$, and we want to find the symmetric functions of the products of roots to get the quartic $x^4+ex^3+fx^2+gx+h$.
First, the constant term $h=(r_1s_1)(r_1s_2)(r_2s_1)(r_2s_2)=r_1^2r_2^2s_1^2s_2^2=b^2d^2$.
Second, the $x^3$ term $e=-(r_1s_1+r_1s_2+r_2s_1+r_2s_2)=-(r_1+r_2)(s_1+s_2)=-ac$.
Third, the $x$ term
begin{align*}g &=-r_1s_1r_1s_2r_2s_1-r_1s_1r_1s_2r_2s_2-r_1s_1r_2s_1r_2s_2-r_1s_2r_2s_1r_2s_2\
&=-r_1r_2s_1s_2(r_1s_1+r_1s_2+r_2s_1+r_2s_2)\
&= -abcdend{align*}
Finally, the $x^2$ term
begin{align*}f &= r_1s_1r_1s_2+r_1s_1r_2s_1+r_1s_1r_2s_2+r_1s_2r_2s_1+r_1s_2r_2s_2+r_2s_1r_2s_2\
&= (r_1^2+r_2^2)s_1s_2+r_1r_2(s_1^2+s_2^2)+2r_1r_2s_1s_2\
&= (a^2-2b)d+(c^2-2d)b+2bd = a^2d+c^2b-2bdend{align*}
Putting those together, the final polynomial is
$$x^4 -ac x^3 + (a^2d+c^2b-2bd)x^2 -abcd x + b^2d^2$$
If we tried to do the same with fewer of the products, to build a quadratic, it just wouldn't work. The building blocks we have are the symmetric polynomials in $r_1,r_2$ and $s_1,s_2$, so anything we build with them will have to be symmetric as well; if $r_1s_1$ is a root, we can exchange $r_1$ and $r_2$ to get $r_2s_1$ as another root, or exchange $s_1$ and $s_2$ to get $r_1s_2$ as a third root, or exchange both to get $r_2s_2$ as a fourth root. It's all four or none. We've shown that getting just those four roots is possible, so that's the best that can be done.
In fact, it's typical that for two irreducible quadratics, the quartic we build this way will be irreducible as well. We'd need a coincidence like the discriminants being the same (up to multiplication by a perfect square) to avoid it.
add a comment |
There is a generic polynomial that will get you the products of the roots - but you can't keep the degree the same. After all, there's not necessarily any way to distinguish between the roots, so if one product appears as a root, all possible products of roots do. For two quadratics, that means that we're looking at a fourth-degree polynomial.
For simplicity, I'll work with two monic quadratics, $x^2+ax+b$ and $x^2+cx+d$, with roots $r_1,r_2$ and $s_1,s_2$ respectively. We have $a=-(r_1+r_2)$, $b=r_1r_2$, $c=(-s_1+s_2)$, $d=s_1s_2$, and we want to find the symmetric functions of the products of roots to get the quartic $x^4+ex^3+fx^2+gx+h$.
First, the constant term $h=(r_1s_1)(r_1s_2)(r_2s_1)(r_2s_2)=r_1^2r_2^2s_1^2s_2^2=b^2d^2$.
Second, the $x^3$ term $e=-(r_1s_1+r_1s_2+r_2s_1+r_2s_2)=-(r_1+r_2)(s_1+s_2)=-ac$.
Third, the $x$ term
begin{align*}g &=-r_1s_1r_1s_2r_2s_1-r_1s_1r_1s_2r_2s_2-r_1s_1r_2s_1r_2s_2-r_1s_2r_2s_1r_2s_2\
&=-r_1r_2s_1s_2(r_1s_1+r_1s_2+r_2s_1+r_2s_2)\
&= -abcdend{align*}
Finally, the $x^2$ term
begin{align*}f &= r_1s_1r_1s_2+r_1s_1r_2s_1+r_1s_1r_2s_2+r_1s_2r_2s_1+r_1s_2r_2s_2+r_2s_1r_2s_2\
&= (r_1^2+r_2^2)s_1s_2+r_1r_2(s_1^2+s_2^2)+2r_1r_2s_1s_2\
&= (a^2-2b)d+(c^2-2d)b+2bd = a^2d+c^2b-2bdend{align*}
Putting those together, the final polynomial is
$$x^4 -ac x^3 + (a^2d+c^2b-2bd)x^2 -abcd x + b^2d^2$$
If we tried to do the same with fewer of the products, to build a quadratic, it just wouldn't work. The building blocks we have are the symmetric polynomials in $r_1,r_2$ and $s_1,s_2$, so anything we build with them will have to be symmetric as well; if $r_1s_1$ is a root, we can exchange $r_1$ and $r_2$ to get $r_2s_1$ as another root, or exchange $s_1$ and $s_2$ to get $r_1s_2$ as a third root, or exchange both to get $r_2s_2$ as a fourth root. It's all four or none. We've shown that getting just those four roots is possible, so that's the best that can be done.
In fact, it's typical that for two irreducible quadratics, the quartic we build this way will be irreducible as well. We'd need a coincidence like the discriminants being the same (up to multiplication by a perfect square) to avoid it.
add a comment |
There is a generic polynomial that will get you the products of the roots - but you can't keep the degree the same. After all, there's not necessarily any way to distinguish between the roots, so if one product appears as a root, all possible products of roots do. For two quadratics, that means that we're looking at a fourth-degree polynomial.
For simplicity, I'll work with two monic quadratics, $x^2+ax+b$ and $x^2+cx+d$, with roots $r_1,r_2$ and $s_1,s_2$ respectively. We have $a=-(r_1+r_2)$, $b=r_1r_2$, $c=(-s_1+s_2)$, $d=s_1s_2$, and we want to find the symmetric functions of the products of roots to get the quartic $x^4+ex^3+fx^2+gx+h$.
First, the constant term $h=(r_1s_1)(r_1s_2)(r_2s_1)(r_2s_2)=r_1^2r_2^2s_1^2s_2^2=b^2d^2$.
Second, the $x^3$ term $e=-(r_1s_1+r_1s_2+r_2s_1+r_2s_2)=-(r_1+r_2)(s_1+s_2)=-ac$.
Third, the $x$ term
begin{align*}g &=-r_1s_1r_1s_2r_2s_1-r_1s_1r_1s_2r_2s_2-r_1s_1r_2s_1r_2s_2-r_1s_2r_2s_1r_2s_2\
&=-r_1r_2s_1s_2(r_1s_1+r_1s_2+r_2s_1+r_2s_2)\
&= -abcdend{align*}
Finally, the $x^2$ term
begin{align*}f &= r_1s_1r_1s_2+r_1s_1r_2s_1+r_1s_1r_2s_2+r_1s_2r_2s_1+r_1s_2r_2s_2+r_2s_1r_2s_2\
&= (r_1^2+r_2^2)s_1s_2+r_1r_2(s_1^2+s_2^2)+2r_1r_2s_1s_2\
&= (a^2-2b)d+(c^2-2d)b+2bd = a^2d+c^2b-2bdend{align*}
Putting those together, the final polynomial is
$$x^4 -ac x^3 + (a^2d+c^2b-2bd)x^2 -abcd x + b^2d^2$$
If we tried to do the same with fewer of the products, to build a quadratic, it just wouldn't work. The building blocks we have are the symmetric polynomials in $r_1,r_2$ and $s_1,s_2$, so anything we build with them will have to be symmetric as well; if $r_1s_1$ is a root, we can exchange $r_1$ and $r_2$ to get $r_2s_1$ as another root, or exchange $s_1$ and $s_2$ to get $r_1s_2$ as a third root, or exchange both to get $r_2s_2$ as a fourth root. It's all four or none. We've shown that getting just those four roots is possible, so that's the best that can be done.
In fact, it's typical that for two irreducible quadratics, the quartic we build this way will be irreducible as well. We'd need a coincidence like the discriminants being the same (up to multiplication by a perfect square) to avoid it.
There is a generic polynomial that will get you the products of the roots - but you can't keep the degree the same. After all, there's not necessarily any way to distinguish between the roots, so if one product appears as a root, all possible products of roots do. For two quadratics, that means that we're looking at a fourth-degree polynomial.
For simplicity, I'll work with two monic quadratics, $x^2+ax+b$ and $x^2+cx+d$, with roots $r_1,r_2$ and $s_1,s_2$ respectively. We have $a=-(r_1+r_2)$, $b=r_1r_2$, $c=(-s_1+s_2)$, $d=s_1s_2$, and we want to find the symmetric functions of the products of roots to get the quartic $x^4+ex^3+fx^2+gx+h$.
First, the constant term $h=(r_1s_1)(r_1s_2)(r_2s_1)(r_2s_2)=r_1^2r_2^2s_1^2s_2^2=b^2d^2$.
Second, the $x^3$ term $e=-(r_1s_1+r_1s_2+r_2s_1+r_2s_2)=-(r_1+r_2)(s_1+s_2)=-ac$.
Third, the $x$ term
begin{align*}g &=-r_1s_1r_1s_2r_2s_1-r_1s_1r_1s_2r_2s_2-r_1s_1r_2s_1r_2s_2-r_1s_2r_2s_1r_2s_2\
&=-r_1r_2s_1s_2(r_1s_1+r_1s_2+r_2s_1+r_2s_2)\
&= -abcdend{align*}
Finally, the $x^2$ term
begin{align*}f &= r_1s_1r_1s_2+r_1s_1r_2s_1+r_1s_1r_2s_2+r_1s_2r_2s_1+r_1s_2r_2s_2+r_2s_1r_2s_2\
&= (r_1^2+r_2^2)s_1s_2+r_1r_2(s_1^2+s_2^2)+2r_1r_2s_1s_2\
&= (a^2-2b)d+(c^2-2d)b+2bd = a^2d+c^2b-2bdend{align*}
Putting those together, the final polynomial is
$$x^4 -ac x^3 + (a^2d+c^2b-2bd)x^2 -abcd x + b^2d^2$$
If we tried to do the same with fewer of the products, to build a quadratic, it just wouldn't work. The building blocks we have are the symmetric polynomials in $r_1,r_2$ and $s_1,s_2$, so anything we build with them will have to be symmetric as well; if $r_1s_1$ is a root, we can exchange $r_1$ and $r_2$ to get $r_2s_1$ as another root, or exchange $s_1$ and $s_2$ to get $r_1s_2$ as a third root, or exchange both to get $r_2s_2$ as a fourth root. It's all four or none. We've shown that getting just those four roots is possible, so that's the best that can be done.
In fact, it's typical that for two irreducible quadratics, the quartic we build this way will be irreducible as well. We'd need a coincidence like the discriminants being the same (up to multiplication by a perfect square) to avoid it.
answered Dec 13 at 6:03
jmerry
1,14616
1,14616
add a comment |
add a comment |
Here's one way to understand why such $g,h,j$ don't necessarily exist:
The roots of an integer quadratic are expressible as
$$frac{xpm sqrt{y}}{z}$$
for some integers $x,y,z$ (say, by the quadratic formula). If you multiply two of these together, you can get something like this:
$$left(frac{x_1+sqrt{y_1}}{z_1}right)left(frac{x_2+sqrt{y_2}}{z_2}right)=frac{x_1x_2+x_1sqrt{y_2}+x_2sqrt{y_1}+sqrt{y_1y_2}}{z_1z_2}.$$
We need these three square-root terms to somehow collapse into one for this to be expressible as $frac{X+sqrt{Y}}{Z}$ for some integers $X,Y,Z$. If there are no nontrivial relations between $sqrt{y_1}$ and $sqrt{y_2}$ (say, if $y_1=2$ and $y_2=3$), then we can't simplify the above at all, and thus it isn't a root of an integer-coefficient quadratic.
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
add a comment |
Here's one way to understand why such $g,h,j$ don't necessarily exist:
The roots of an integer quadratic are expressible as
$$frac{xpm sqrt{y}}{z}$$
for some integers $x,y,z$ (say, by the quadratic formula). If you multiply two of these together, you can get something like this:
$$left(frac{x_1+sqrt{y_1}}{z_1}right)left(frac{x_2+sqrt{y_2}}{z_2}right)=frac{x_1x_2+x_1sqrt{y_2}+x_2sqrt{y_1}+sqrt{y_1y_2}}{z_1z_2}.$$
We need these three square-root terms to somehow collapse into one for this to be expressible as $frac{X+sqrt{Y}}{Z}$ for some integers $X,Y,Z$. If there are no nontrivial relations between $sqrt{y_1}$ and $sqrt{y_2}$ (say, if $y_1=2$ and $y_2=3$), then we can't simplify the above at all, and thus it isn't a root of an integer-coefficient quadratic.
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
add a comment |
Here's one way to understand why such $g,h,j$ don't necessarily exist:
The roots of an integer quadratic are expressible as
$$frac{xpm sqrt{y}}{z}$$
for some integers $x,y,z$ (say, by the quadratic formula). If you multiply two of these together, you can get something like this:
$$left(frac{x_1+sqrt{y_1}}{z_1}right)left(frac{x_2+sqrt{y_2}}{z_2}right)=frac{x_1x_2+x_1sqrt{y_2}+x_2sqrt{y_1}+sqrt{y_1y_2}}{z_1z_2}.$$
We need these three square-root terms to somehow collapse into one for this to be expressible as $frac{X+sqrt{Y}}{Z}$ for some integers $X,Y,Z$. If there are no nontrivial relations between $sqrt{y_1}$ and $sqrt{y_2}$ (say, if $y_1=2$ and $y_2=3$), then we can't simplify the above at all, and thus it isn't a root of an integer-coefficient quadratic.
Here's one way to understand why such $g,h,j$ don't necessarily exist:
The roots of an integer quadratic are expressible as
$$frac{xpm sqrt{y}}{z}$$
for some integers $x,y,z$ (say, by the quadratic formula). If you multiply two of these together, you can get something like this:
$$left(frac{x_1+sqrt{y_1}}{z_1}right)left(frac{x_2+sqrt{y_2}}{z_2}right)=frac{x_1x_2+x_1sqrt{y_2}+x_2sqrt{y_1}+sqrt{y_1y_2}}{z_1z_2}.$$
We need these three square-root terms to somehow collapse into one for this to be expressible as $frac{X+sqrt{Y}}{Z}$ for some integers $X,Y,Z$. If there are no nontrivial relations between $sqrt{y_1}$ and $sqrt{y_2}$ (say, if $y_1=2$ and $y_2=3$), then we can't simplify the above at all, and thus it isn't a root of an integer-coefficient quadratic.
edited Dec 13 at 17:08
answered Dec 13 at 5:20
Carl Schildkraut
11.1k11441
11.1k11441
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
add a comment |
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
@ruakh Thanks -- fixed now.
– Carl Schildkraut
Dec 13 at 17:08
add a comment |
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That's impossible. Consider $x^2+2x-4=0$ and $x^2+4x-4=0$, the minimal polynomial of the product of the roots of equation is quartic.
– Kemono Chen
Dec 13 at 5:07
@KemonoChen which pair of roots?
– The Great Duck
Dec 13 at 5:10
Each pair of roots is OK.
– Kemono Chen
Dec 13 at 5:11
@KemonoChen im confused. You said some root of the first equation times some root of the second equation has a quartic as its minimal polynomial. Are you saying they are all like that or are you saying any two will not require a quartic?
– The Great Duck
Dec 13 at 5:12
Yes. ${}{}{}{}$
– Kemono Chen
Dec 13 at 5:13