Difference between revisions of "Cohn's criterion"
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The following proof is due to M. Ram Murty. | The following proof is due to M. Ram Murty. | ||
− | We start off with a lemma. Let <math>g(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]</math>. Suppose <math>a_n\geq 1</math>, | + | We start off with a lemma. Let <math>g(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]</math>. Suppose <math>a_n\geq 1</math>, <math>|a_i|\leq H</math>. |
+ | Then, any complex root of <math>g(x)</math>, <math>\phi</math>, has a non positive real part or satisfies <math>|\phi|<\frac{1+\sqrt{1+4H}}{2}</math>. | ||
Proof: If <math>|z|>1</math> and Re <math>z>0</math>, note that: | Proof: If <math>|z|>1</math> and Re <math>z>0</math>, note that: | ||
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This means <math>g(z)>0</math> if <math>|z|\geq \frac{1+\sqrt{1+4H}}{2}</math>, so <math>|\phi|<\frac{1+\sqrt{1+4H}}{2}</math>. | This means <math>g(z)>0</math> if <math>|z|\geq \frac{1+\sqrt{1+4H}}{2}</math>, so <math>|\phi|<\frac{1+\sqrt{1+4H}}{2}</math>. | ||
− | If <math>b\geq 3</math>, this implies <math>|b-\phi|>1</math> if <math>b\geq 3</math> and <math>f(\phi)=0</math>. Let <math>f(x)=g(x)h(x)</math>. Since <math>f(b)=p</math>, one of <math>|g(b)|</math> and <math>h(b)</math> is 1. WLOG, assume <math>g(b)=1</math>. Let <math>\phi_1, phi_2,\cdots,\phi_r</math> be the roots of <math>g(x)</math>. This means that <math>|g(b)|=|b-\phi_1||b-\phi_2|\cdots|b-\phi_r|>1</math>. Therefore, <math>f(x)</math> is irreducible. | + | If <math>b\geq 3</math>, this implies <math>|b-\phi|>1</math> if <math>b\geq 3</math> and <math>f(\phi)=0</math>. Let <math>f(x)=g(x)h(x)</math>. Since <math>f(b)=p</math>, one of <math>|g(b)|</math> and <math>h(b)</math> is 1. WLOG, assume <math>g(b)=1</math>. Let <math>\phi_1, \phi_2,\cdots,\phi_r</math> be the roots of <math>g(x)</math>. This means that <math>|g(b)|=|b-\phi_1||b-\phi_2|\cdots|b-\phi_r|>1</math>. Therefore, <math>f(x)</math> is irreducible. |
If <math>b=2</math>, we will need to prove another lemma: | If <math>b=2</math>, we will need to prove another lemma: | ||
− | All of the zeroes of <math>f(x)</math> satisfy Re <math>z | + | All of the zeroes of <math>f(x)</math> satisfy Re <math>z<\frac{3}{2}</math>. |
Proof: If <math>n=1</math>, then the two polynomials are <math>x</math> and <math>x\pm 1</math>, both of which satisfy our constraint. For <math>n=2</math>, we get the polynomials <math>x^2</math>, <math>x^2\pm x</math>, <math>x^2\pm 1</math>, and <math>x^2\pm x\pm 1</math>, all of which satisfy the constraint. If <math>n\geq 3</math>, | Proof: If <math>n=1</math>, then the two polynomials are <math>x</math> and <math>x\pm 1</math>, both of which satisfy our constraint. For <math>n=2</math>, we get the polynomials <math>x^2</math>, <math>x^2\pm x</math>, <math>x^2\pm 1</math>, and <math>x^2\pm x\pm 1</math>, all of which satisfy the constraint. If <math>n\geq 3</math>, | ||
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For <math>|z|\geq \frac{3}{2}</math>, then <math>|z|^2(|z|-1)\geq(\frac{3}{2})^2(\frac{1}{2})=\frac{9}{8}>1</math>. Therefore, <math>z</math> is not a root of <math>f(x)</math>. | For <math>|z|\geq \frac{3}{2}</math>, then <math>|z|^2(|z|-1)\geq(\frac{3}{2})^2(\frac{1}{2})=\frac{9}{8}>1</math>. Therefore, <math>z</math> is not a root of <math>f(x)</math>. | ||
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− | To finish the proof, let <math>f(x)=g(x)h(x)</math>. Since <math>f(2)=p</math>, one of <math>g(2)</math> and <math>h(2)</math> is 1. WLOG, assume <math>g(2)=1</math>. By our lemma, <math>|z-2|>|z-1|</math>. Thus, if <math>\phi_1, \phi_2,\cdots,\phi_r</math> are the roots of <math>g(x)</math>, then <math>|g(2)|=|2-\phi_1||2-\phi_2|\cdots|2-\phi_n|>|1-\phi_1||1-\phi_2|\cdots|1-\phi_n|=|g(1)| | + | To finish the proof, let <math>f(x)=g(x)h(x)</math>. Since <math>f(2)=p</math>, one of <math>g(2)</math> and <math>h(2)</math> is 1. WLOG, assume <math>g(2)=1</math>. By our lemma, <math>|z-2|>|z-1|</math>. Thus, if <math>\phi_1, \phi_2,\cdots,\phi_r</math> are the roots of <math>g(x)</math>, then <math>|g(2)|=|2-\phi_1||2-\phi_2|\cdots|2-\phi_n|>|1-\phi_1||1-\phi_2|\cdots|1-\phi_n|=|g(1)| < 1</math>. This is a contradiction, so <math>f(x)</math> is irreducible. |
[[Category:Algebra]] | [[Category:Algebra]] |
Latest revision as of 07:37, 4 March 2021
Let be a prime number, and an integer. If is the base- representation of , and , then is irreducible.
Proof
The following proof is due to M. Ram Murty.
We start off with a lemma. Let . Suppose , . Then, any complex root of , , has a non positive real part or satisfies .
Proof: If and Re , note that: This means if , so .
If , this implies if and . Let . Since , one of and is 1. WLOG, assume . Let be the roots of . This means that . Therefore, is irreducible.
If , we will need to prove another lemma:
All of the zeroes of satisfy Re .
Proof: If , then the two polynomials are and , both of which satisfy our constraint. For , we get the polynomials , , , and , all of which satisfy the constraint. If ,
If Re , we have Re , and then For , then . Therefore, is not a root of .
To finish the proof, let . Since , one of and is 1. WLOG, assume . By our lemma, . Thus, if are the roots of , then . This is a contradiction, so is irreducible.