Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 360728 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0487 1209376 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0487 1209376 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1262 4189832 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1276 4217432 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1284 4235144 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1286 4265664 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1296 4288384 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1296 4288384 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1821 4966072 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1858 5158224 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1979 5179152 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2093 5376176 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2093 5376432 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360728 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0487 1209376 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0487 1209376 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1262 4189832 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1276 4217432 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1284 4235144 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1286 4265664 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1296 4288384 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1296 4288384 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1821 4966072 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1858 5158224 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1979 5179152 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3548 13807208 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.3550 13807608 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.3550 13807608 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.3552 13815856 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.3555 13817592 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.3561 13850640 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.3561 13851040 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.3562 13851120 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Gauss's Lemma

Gauss's Lemma
Theorem: Let  
\[f \in \mathbb{Z}[x]\]
 . Then  
\[f\]
  is irreducible over  
\[\mathbb{Z}[x]\]
  if and only if  
\[f\]
  is irreducible over  
\[\mathbb{Q}[x]\]
 .
(In other words, Let  
\[f(x)\]
  be a polynomial with integer coefficients. If  
\[f(x)\]
  has no factors with integer coefficients, then 
\[f(x)\]
  has no factors with rational coefficients.)
Proof: Let 
\[f(x) = g(x)h(x)\]
  be a factorization of 
\[f\]
  into polynomials with rational coefficients. Then for some rational 
\[a\]
  the polynomial 
\[a g(x)\]
  has integer coefficients with no common factor. Similary we can find a rational 
\[b\]
  so that 
\[b h(x)\]
  has the same properties. (Take the lcm of the denominators of the coefficients in each case, and then divide by any common factors.)
Suppose a prime 
\[p\]
  divides 
\[a b\]
 . Since  
\[ a b f(x) = (a g(x))(b h(x)) \]
 
becomes 
\[ 0 = (a g(x)) (b h(x))\]
  modulo 
\[p\]
 , we see 
\[a g(x)\]
  or 
\[b h(x)\]
  is the zero polynomial modulo 
\[p\]
 . (If not, then let the term of highest degree in 
\[a g(x)\]
  be 
\[m x^r\]
 , and the term of highest degree in 
\[b h(x)\]
  be 
\[n x^s\]
 . Then the product contains the term 
\[m n x^{r+s} \ne 0 \pmod {p}\]
 , a contradiction.)
In other words, 
\[p\]
  divides each coefficient of 
\[a g(x)\]
  or 
\[b h(x)\]
 , a contradiction. Hence 
\[a b = 1\]
  and we have a factorization over the integers.
Example: Let 
\[p\]
  be a prime. Consider the polynomial  
\[ f(x) = 1 + x + ... + x^{p-1} . \]

We cannot yet apply the criterion, so make the variable subsitution 
\[x = y + 1\]
 . Then we have  
\[ g(y) = 1 + (y+1) + ... + (y+1)^{p-1} . \]

Note 
\[f(x)\]
  is irreducible if and only if 
\[g(y)\]
  is irreducible.
The coefficient of 
\[y^k\]
  in 
\[g(y)\]
  is  
\[ \sum_{m=k}^{p-1} {\binom{p-1}{k}} = {\binom{p}{k+1}} . \]

The last equality can be shown via repeated applications of Pascal’s identity:  
\[ {\binom{n+1}{k}} = {\binom{n}{k}} + {\binom{n}{k-1}} . \]
 
Alternatively, use the fact  
\[ g(y) = \frac{(y+1)^p - 1}{(y+1) - 1} \]
 
Thus 
\[p\]
  divides each coefficient except the leading coefficient, and 
\[p^2\]
  does not divide the constant term 
\[p\]
 , hence 
\[f(x)\]
  is irreducible over the rationals.

Add comment

Security code
Refresh