Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where   $n$ is the degree of $f$, $n&gt;2$. If $\alpha$ is a root of $f$ in the splitting field, then we have to show that  $\mathbb{Q}(\alpha^4)=\mathbb{Q}(\alpha)$.  My attempt: $\mathbb{Q}(\alpha^4)\subseteq\mathbb{Q}(\alpha)$ is obvious.Also, $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^4)]=4$ as $x^4-\alpha^4$ is the minimal polynomial of $\alpha$ over $\mathbb{Q}(\alpha^4)$. So, $n$ must be a multiple of $4$, by Tower Law.  I have come to a dead end it seems Please guide me.
From Berkeley Problems in Mathematics, Spring 1999, .  Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\ge 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in L$ be a zero of $f$. Give that $[L:\mathbb{Q}]=n!$, prove that $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$.   (BELOW IS NONSENSE, SHOULD BE IGNORED)  Following Gerry Myerson's advice, since $L$ is the splitting field of $f$, $L$ must be Galois over $\mathbb{Q}$. Thus the Galois group for $[L:\mathbb{Q}]=S_{n}$. Gerry now asserts that either $\mathbb{Q}(\alpha^{4})$ is of degree 2 over rationals or it must be $\mathbb{Q}$. But assuming this one can rule out the case $\mathbb{Q}$, since this would imply $\alpha$ is the root of a 4th or 2nd degree polynomial over $\mathbb{Q}$. But we know $n\ge 3$. So $\alpha$ must be the root of a 4th degree polynomial. We know $S_{n}$ has normal subgroups $A_{n}$ when $n\not=4$ and $V,A_{4}$ when $n=4$. I do not know how to proceed any further.   Now assuming $\mathbb{Q}(\alpha^{4})$ is a degree 2 abelian extension over $\mathbb{Q}$. We should have a chain of normal extensions $\mathbb{Q}(\alpha)\supset \mathbb{Q}(\alpha^{2})\supset \mathbb{Q}$. This would imply $S_{n}$ has a normal subgroup which has a normal subgroup. But we know $A_{n}$ for $n\ge 5$ are simple. So the only possibility is $\mathbb{Q}(\alpha)$ has a Galois group isomorphic to $V$. In this case $n=4$ as well. I can only proceed to here.
Round function in Excel, worksheet function vs VBA Does "notified by [date]" include the end date? I have read the Rules of a competition. The text of the Rules include a sentence as follows:     As per stated in the Rules the entrants will be notified by May 30th 2010.   Does the sentence above mean that May 30th is included in the term of notification or not? Definition of "pepper" in hash functions When transiting between two flights, will my luggage be automatically transferred? Maximum Champion Growth size I was reading about Lulu (new champion) and saw this:     Wild Growth (Ultimate) - Lulu enlarges an ally, knocking enemies away   from them and granting them a large amount of bonus health. For the   next few seconds, that ally gains an aura that slows nearby enemies.   Then I start wondering: what is the max size that a champion reach ? Is there a Maximum size ?  Example: Cho'Gath's Feast +  +  + Wild Growth + Something that make champions grow in size. Cho'Gath will grow every time ? or it will stop at some point ? Why should we care about groups at all? WhatsApp for Ubuntu Phone? Does someone know if there already is a WhatsApp Version for Ubuntu Phone? I decided to install Ubuntu on my Android, but I don't want to use my phone without WhatsApp. What does "user was removed" mean and why did my reputation change because of it? Message appearing when a user was removed.     It seems I lost 1765 reputation points on Stack Overflow yesterday. The reason given in my history is "removed", with the description of "User was removed ()". I don't understand what that means. What caused this? Why would you not want to rollover a previous employer's 401(k) when changing jobs? Are there any valid reasons to not rollover a former 401(k) when changing jobs?  I have several little ones that I never quite got around to rolling together - with an upcoming job change, should they all be combined into my new employer's retirement plan?  If not, what else might be considered? What is the advantage of using wp_mail? What is the advantage of using wp_mail() over mail(). Codex says they're similar, but they seem to be very similar. MacBook is displaying the message "no hardware installed" when I click on Wi-Fi I have a MacBook Pro 2009 running MacOS El Capitan. Recently the Wi-Fi has stopped working.   When I click on the Wi-Fi icon it displays the message "No hardware installed", but Ethernet still works.   Is there any way of fixing this by using a Wi-Fi dongle or something of the sort, if so do I need a specific dongle and how do I install it? Illustrator: Is there an efficient and accurate way to join two circles by their tangents (and keep it editable)? It's easy to join two circles if they are the same size and horizontal/vertical, into a pill shape:    If the circles are at an add angle then it's still possible but you can no longer rely on smart guides.     You can add lines roughly where the tangents should be and then use the scissors tool on the circles, cut out the inner area and then join the two crescents into one shape.    Is there a better way to do this? Is there a way to do this that is still editable afterwards too?  There are some other questions that cover the same area as this but don't cover editability or joining circles that are not in line with each other (where smart guides can't help). How do we know Dark Matter isn't simply Neutrinos? Auto-expand TAB character in code formatting Related:       Note that this question, while related, is not the same as the ones linked.  In many cases, when people put code into questions or answers, they copy/paste the code from their editor.  If their editor is using tab characters, the formatting will be broken after the post is submitted.  For example, if a code is indented using TAB characters, then one TAB character in the beginning of a line will be treated as one level of indent and align with the left edge of the code, effectively indenting equally to 4 spaces.  However in the post editor it looks indented twice - the same as 8 spaces.  I often edit other people's posts to improve code formatting and I find this TAB issue being the biggest annoyance.  Most of the times (I think) on SO, we use 4 spaces to format/indent the code.  I think it would be an easy thing to auto-expand TAB characters inside &lt;code&gt; blocks by replacing them with 4 spaces when the post is submitted. Regression - How do I know if my residuals are normally distributed? Performing a regression and need to find out if my residuals are normally distributed. What is the difference between Parse() and TryParse()?  int number = int.Parse(textBoxNumber.Text);  // The Try-Parse Method int.TryParse(textBoxNumber.Text, out number);   Is there some form of error-checking like a Try-Catch Block? lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. Why do I have the "required reviews completed" status on my submission in Elsevier? I had submitted my paper in an Elsevier journal one year back. Now, status update on the submission tracking says "Required reviews completed." However, for the last two weeks still there is no change in the status.  Should I ask the editor what is going on with my submission? Is there an easy way of getting the charset and collation of the DB tables in WordPress without resorting to SQL queries? How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$?