I'm looking for good books of abstract algebra. In particular, I need some books with a lot of exercises, solved and not (in particular about modular arithmetic, groups and rings) Do you have any hints? Thanks.
Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study friendly (good syllabus overall though).  The course was split up into 3 parts, group theory, ring theory and field theory. As a computer science major we only had to study the first 2.  Now that I passed the exam for this course I want to study the field theory part ( which covers Galois theory, etc).  So, now I want to ask whether any of you know any good books on abstract algebra, which lift off at basic ring theory and continue to more advanced ring theory and to finite fields, Galois theory, ...  Please keep in mind that I am not a math major, and that I would like books which are suited for self study (thus a lot of examples and intuition).  Thanks in advance!
Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$. Find the minimum value of $f(x,y,z)=\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+z)(y+x)}+\frac{z^2}{(z+x)(z+y)}$ for all notnegative value of $x,y,z$.  I think that minimum value is $\frac{3}{4}$ when $x=y=z$ but I have no prove. 12.04 GPU lockup I was able to install Ubuntu 12.04 using the alternative version of the image, (64 bit) but when I start the system it tells me the GPU  locked up, and instead presents a text only screen.  I understand that this site is not for bug reports, but at the same time, I know that there are some versions of the installer than have to be compatible with my graphics card (Nvidia GTS 360M).  I assumed that the mere fact that the alternative installer ran was an indication that I could get some sort of graphical interface running after the install.  Is there some way of getting some type of graphical interface running on my machine? Even better would be is there a way to get the rest of the system running as normal (i.e. GPU up and running)?  If an internet connection is required to download driver support or something similar, directions for how to configure a wireless connection with WPA2-Personal protection from the text-only interface (bash).  Any help would be appreciated.  Note: I was able to find  which tells me that the card is supported.  Note 2: In the install software stage of the install, I was unable to install anything (no internet connection)  Update: The exact error message is [8.591394] [drm] 0000:01:00.0: GPU Lockup - switching to software fbcon  Update 2: If I try to install with the regular version of Ubuntu, the screen goes staticy. This also happens if I install with Wubi (distasteful I know) and then try to use Ubuntu.  Update 3: I've tried xforcevesa, nomodeset and xforcevesa nomodeset as my wubi boot options.  Update 4: Thanks to jokerdino, I can now connect to the internet, so internet tools would now be a possibility. Why don't we consider centrifugal force on a mass placed on earth? Creating polygon feature class from area in one not covered by another using ArcGIS Desktop? I am working on a project doing spatial analysis at a statewide level. I am looking at a form of transportation coverage in a state, and part of my analysis involves buffers and drive time polygons. I am using a geodatabase (therefore ArcGIS) for all my work, so new features will be exported as feature classes in feature datasets. To visualize: I have said state border, and some dissolved buffer rings around some points, and they cover parts of the state. I want to create a feature class for the parts of the state these buffer rings don't cover, AKA gap areas. Is there a tool I can use or create to do this? I know I could draw them in manually in editor, but I am unsure if that's the most precise way to generate these feature classes. To me, I am thinking this operation would be like an inverted &quot;Clip&quot; in that I take the two layers, and instead of pulling out the clipped area, it would take everything not in the clipped area. I am using a Desktop Basic license. Responsive Breakpoints? Twitter Bootstrap uses these break points for responsive design:  Large Display: 1200px Default: 980px Tablet: 768px Mobile: 480px  A client said they are using 1280px for their Large Display breakpoint.  Is there an industry standard for these or are they just made up as we go right now? Why does the reputation dropdown not show this upvote? What is serial voting and how does it affect me? I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? Is there a way to find out if I need a transit visa for a layover in the UK? Is there a website or some other way to find out if I need a transit visa for a short stop in the UK on my way somewhere else? Ideally this should give me the answer for all nationalities, and take into account any other visas I may have. I just upgraded linux-firmware package and after that I used $ sudo update-initramfs -u -k all but I got some errors:  $ sudo update-initramfs -u -k all [sudo] password for piotrek:  update-initramfs: Generating /boot/initrd.img-4.4.0-34-generic Warning: No support for locale: en_US.utf8 update-initramfs: Generating /boot/initrd.img-4.4.0-31-generic Warning: No support for locale: en_US.utf8 update-initramfs: Generating /boot/initrd.img-4.4.0-13-generic WARNING: missing /lib/modules/4.4.0-13-generic Ensure all necessary drivers are built into the linux image! depmod: ERROR: could not open directory /lib/modules/4.4.0-13-generic: No such file or directory depmod: FATAL: could not search modules: No such file or directory Warning: No support for locale: en_US.utf8 depmod: WARNING: could not open /var/tmp/mkinitramfs_LJbkti/lib/modules/4.4.0-13-generic/modules.order: No such file or directory depmod: WARNING: could not open /var/tmp/mkinitramfs_LJbkti/lib/modules/4.4.0-13-generic/modules.builtin: No such file or directory   I'm sure I don't have kernel 4.4.0-13. I might have had it some time ago but removed it:   $ apt list --installed | grep linux-  WARNING: apt does not have a stable CLI interface. Use with caution in scripts.  linux-base/xenial,xenial,now 4.0ubuntu1 all [installed,automatic] linux-firmware/now 1.159 all [installed,local] linux-headers-4.4.0-31/xenial-updates,xenial-updates,xenial-security,xenial-security,now 4.4.0-31.50 all [installed] linux-headers-4.4.0-31-generic/xenial-updates,xenial-security,now 4.4.0-31.50 amd64 [installed] linux-headers-4.4.0-34/xenial-updates,xenial-updates,xenial-security,xenial-security,now 4.4.0-34.53 all [installed] linux-headers-4.4.0-34-generic/xenial-updates,xenial-security,now 4.4.0-34.53 amd64 [installed] linux-image-4.4.0-31-generic/xenial-updates,xenial-security,now 4.4.0-31.50 amd64 [installed] linux-image-4.4.0-34-generic/xenial-updates,xenial-security,now 4.4.0-34.53 amd64 [installed] linux-image-extra-4.4.0-31-generic/xenial-updates,xenial-security,now 4.4.0-31.50 amd64 [installed] linux-image-extra-4.4.0-34-generic/xenial-updates,xenial-security,now 4.4.0-34.53 amd64 [installed] linux-libc-dev/xenial-updates,xenial-security,now 4.4.0-34.53 amd64 [installed] linux-sound-base/xenial,xenial,now 1.0.25+dfsg-0ubuntu5 all [installed] syslinux-common/xenial,xenial,now 3:6.03+dfsg-11ubuntu1 all [installed] syslinux-legacy/xenial,now 2:3.63+dfsg-2ubuntu8 amd64 [installed]   How to fix those update-initramfs errors? jQuery get all divs which do not have class attribute public static map&lt;id,RecordType&gt; MapRecordtype = new Map&lt;id,RecordType&gt;([SELECT id, DeveloperName FROM RecordType where sobjecttype ='move__c' and                                         Developername in('Assignment','Draft','Domestic_Draft','LTST_Draft','MX_Domestic','MX_Domestic_draft','Termination','International_Relo','Localization','J1',                                        'Back_To_Back','Repatriation','PAM','US_Domestic_Relo','CHN_Domestic_Relo','CHN_Domestic_Draft')])   Can we write schema for above query? XSL That Returns the XML unchanged is it problematic to cluster across two datacenters w/aag? In general is it problematic to cluster across two datacenters that are up to 15-miles apart?  For this one particular project the approach is the following?   2-Node Failover Cluster across two datacenters, one node per datacenter local SAN disks, no shared disks 'Node and FileShare Majority' at third location on-campus employ AAG, failover will be manual; asynchronous mode   Is anyone implementing something like this with success??!?  Thanks so much for the help!! Algorithm to find the irreducible polynomial What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a field $GF(p^k)$ requires finding a irreducible polynomial. (I'm mostly interested in $GF(2^k)$.) How can I restart an app game from the beginning Do ISTQB/ISEB Testing Certificates prove that someone can test? I have passed the , and to be honest, I'm pretty sure that anyone could pass that exam with a couple of days of studying. My question is are these certificates really of value to my career and should I put any effort into acquiring the ? We have to shift our web-site to a new domain due to some unavoidable circumstances. The portal has been up and running there since 2009.  It has good index in various search engines and we are getting a good number of page hits on daily basis.  Losing the search hits and index are our big concern. We are not sure how best we can handle this.  What all can be done to preserve as much as possible while shifting to new domain name. The value of a, b and c for this integral Find out the value of $a, b$ and $c$, for which $$ \lim _{x\to 0}\:\:\frac{1}{x^5}\left(\int _0^x\:\left(a+b \cos\left(t\right)+c \cos\left(2t\right)\right)dt\right)=\frac{1}{5} $$  My first thought was to separate the integral into multiple ones because i thought that it is pretty clear that each integral is a common one and a,b,c are constants and can be taken out . After I solved the integral, that's where I had the problem. I couldn't do the limit of the result that I've got in order to find out the value of $a,b$ and $c$ . For which values of $x$, will $f (x) . f '(x) < 0$? Book recommendation on the history of PDE/ODE?