Show that $[\mathbb{Q}(\sqrt[4]2,\sqrt3):\mathbb{Q}]=8$.   Clearly $[\mathbb{Q}(\sqrt[4]2):\mathbb{Q}]=4$, so it suffices to show $\sqrt3\notin\mathbb{Q}(\sqrt[4]2)$. I could only think of setting $(a+b\sqrt[4]2+c\sqrt[4]4+d\sqrt[4]8)^2=(e\sqrt3)^2$ and calculate, but this is tedious. Is there any quick way to do that?
I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$.  This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois Theory. The problem asked to show that $$ \left[ \mathbb{Q}{ \left( \sqrt[4]{2},\sqrt{3} \right):\mathbb{Q} }\right] =8 $$ Here's my work : First, we have $$  \left[ \mathbb{Q} ( \sqrt[4]{2},\sqrt{3}):\mathbb{Q}\right] = \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] . \left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right]$$ It's easy to show that $\left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right] =4 $ because the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ is $x^4 -2$. Next, I'm going to show that  $$ \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] =2$$ Consider the polynomial $f(x) = x^2 -3$. It's monic and $\sqrt{3}$ is a root of $f$, but I'm stuck in proving $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ (hence $f$ is irreducible over $\mathbb{Q}(\sqrt[4]{2}))$.  My question is: Is there any proof to show that $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ or another way to caculate the degree of this field extension?  Thank you for your help.
An idempotent operator is compact if and only if it is of finite rank Implementing the decorator pattern in Python Boot Error - ACPI PCC Probe Failed ACPI PCC Probe Failed is the error that pops up instead of the default GRUB loader after I updated my kernel to 3.19.  I searched around and couldn't find much information except that it is a known error and it is registered in kernel.org's launchpad . My question is, is anyone else facing it?, what are the impacts of this? And if only an update in the future can fix it, any suggestions or safety tips till then? $(U\cap W)^0 = U^0 + W^0$ Let $V$ be a finite dimensional vector space and $U^0$ and $W^0$ denote the annihilators of subspaces $U$ and $W$, respectively.  Then I need to show that $(U\cap W)^0 = U^0 + W^0$.  I can show that $U^0 + W^0 \subseteq (U\cap W)^0$ like so: Let $f\in U^0$ and $g\in W^0$.  Then, because $U^0, W^0 \subseteq (U\cap W)^0$, we see that if $f\in U^0$ and $g\in W^0$, then for any $v\in U\cap W$ and $a,b \in \Bbb F$, $(af+bg)(v) = af(v) + bg(v) = 0$.  Thus $af+bg\in (U\cap W)^0$.  But because $f$ and $g$ were arbitrary, this implies that any linear combination of elements of $U^0$ and $W^0$ is an element of $(U\cap W)^0$.  I.e. $U^0+W^0 \subseteq (U\cap W)^0$.  But I haven't had any luck proving the other direction.  Obviously I should start with $f\in (U\cap W)^0$.  But then it looks like I need to somehow break that function into two to show its in $U^0 + W^0$.  That's throwing me off. Is it possible to check where an alias was defined? Using `\input` to insert a section of a document into another I don't see advantages of using events over delegates, other than being syntactical sugar. Perhaps I am misunderstanding, but it seems that event is just a placeholder for delegate.  Would you explain to me the differences and when to use which?  What are the advantages and disadvantages? Our code is heavily rooted with events, and I want to get to the bottom of it.  When would you use delegates over events and vice versa? Please state your real world experience with both, say in the production code. I'm reading . Then I'm not able to figure out the difference between hard voting and soft voting in context to ensemble based methods.  I quote descriptions of them from the book. The first two images from the top are description for hard voting, and last one is for soft voting.        In my view hard voting is majority decision, but I don't understand soft voting and the reason why soft voting is better than hard voting. Would anyone teach me these? How do I define custom web.config sections with potential child elements and attributes for the properties? The web applications I develop often require co-dependent configuration settings and there are also settings that have to change as we move between each of our environments. All our settings are currently simple key-value pairs but it would be useful to create custom config sections so that it is obvious when two values need to change together or when the settings need to change for an environment. What's the best way to create custom config sections and are there any special considerations to make when retrieving the values? What would you call a word that doesn't exist in or translate well into another language? Children's book with "square triangles" in parallel universe? In the 1960s I read book--an older children's book, I think--which I loved, and still remember, roughly.  I have no idea what the title is or who the author is, and I'm wondering if anyone knows this one:  Scientists had invented a device which they didn't fully understand, and which had puzzling effects.  For example, when a pencil was put into the machine, it disappeared, and then reappeared inside out: The lead was on the outside.  When people went into the machine, they came back dead, having died from fright.  Finally they sent a very open-minded person into the machine.  She was an artist, I think.  She was able to return alive, and described an alternate universe in which geometry was very different.  I distinctly remember that there were supposedly square triangles or four-sided triangles--or was it three-sided squares?  (This is, of course, a literally contradictory claim, but I didn't see the matter as clearly then.)  Thanks! ORA-00979 not a group by expression How do I blend multiple photos to remove random people from the background? I'm very excited about the new  for Android/iPhone. Let's say you're taking a picture of your friend in front of a landmark with several people walking by. The app allows you to remove the people from the photo, leaving only the background and your friend. The app isn't just doing a clone of the surrounding image to paint over unwanted subjects. It's actually taking multiple images, detecting which objects are moving, and letting you select them for removal. This sounds awesome! But I don't want to be limited to taking pictures with a smart phone.  I would like to take multiple photos of the same scene. (Maybe with a tripod, but it would be better if I didn't have to use a tripod.) Then I would somehow blend them together to remove unwanted elements. How do I do that? I know there are tons of photoshop tutorials out there, but all the ones I find just tell you how to use the clone tool. Install GNOME Shell extensions from terminal Is there a way to install GNOME extensions from terminal, for example dash to dock? The way I do it now is to go into Ubuntu Software app store and install it. A bunch of CSS files were pulled in and now I'm trying to clean things up a bit. How can I efficiently identify unused CSS definitions in a whole project? Does anyone know how prove the following?   Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. Show that $f$ and $g$ are constant on $\Omega$. Let's assume you have a table groups and a table item. Each item belongs to exactly one group. It is an inherent part of that group. An item cannot exist outside of a group and it cannot be moved to another group.  When trying to decide on a primary key for the item table, what should I use?  Should I make up an artificial global serial key like this:  CREATE TABLE items (     item serial PRIMARY KEY,     group integer NOT NULL REFERENCES groups(group), );   ... or should I rather use a composite key and per-group item serial like this:  CREATE TABLE items (     group integer NOT NULL REFERENCES groups(group),     item integer NOT NULL,      PRIMARY KEY(group, item) );   The reason why I'm leaning more towards the second solution is that the post URL will always show the group and item, so it makes sense to have both of them as the composite primary key. In case of the first solution, the URL contains superfluous information because the group ID can already be deduced from the item ID alone. The URL structure is given, however, and cannot be changed.  The disadvantage of the second solution is that you have to manage a per-group serial (i.e. each item integer should start from 0 for each group).  What's better in terms of best practices, normalization and performance? Or is it simply a matter of taste? Pytest is in PATH but not found inequality with three variables and condition Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number    I have tried to look at $$\frac{z+1}{z-1}=\frac{z+1}{z-1}\cdot \overline{\frac{z-1}{\overline{z-1}}}=\frac{z+1}{z-1}\cdot \frac{\overline{z}-1}{\overline{z}-1}$$  But did not get far