In abstract algebra we define $\mathbb{Q}(\sqrt{2})$ to be the smallest field that contains $\mathbb{Q}$ and $\sqrt{2}$. In the most of textbox they usually write that this is. $$\mathbb{Q}(\sqrt{2}) = \{a+b\sqrt{2}\mid a,b\in \mathbb{Q}\}$$ But can I prove that this is the smallest field that contains $\mathbb{Q}$ and $\sqrt{2}$.?
This was an assertion made in our textbook but I have no idea how to show that either statement is true.  Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, which was the second part of the assertion.
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Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on  seems to agree:     Molecular and quantum physics show that the electromagnetic force is the fundamental interaction responsible for contact forces.    An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what force is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such.  I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. 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Notes:  Generating smaller indices seems to work just fine, only when I attempt to generate the entire index do I run into problems, so coming up with a MWE will be difficult. What are the uses of "using" in C#? User  answered the wonderful  question by mentioning the using keyword. Can you elaborate on that? What are the uses of using? any latex code I want to view in TexWorks 0.4.5 r.1280 (Miktex 2.9) won't show the syntax highlighting. It compiles just fine but everything line is written in black. On very long documents the highlighting shows for several seconds then the program freezes for short and the highlighting disappears again. Windows compatibility modes did not improve the situation. Does anyone have an idea what's going wrong here? How can I tell if a CVE has been fixed in Ubuntu's repositories? Today a couple of buffer overflows in NTP were announced , . It looks like updating my system to fix these issues is in order.   How can I find out if they have been fixed in the Ubuntu repositories, such that if I were to run:   sudo apt-get update sudo apt-get upgrade   then the fix would get installed and the vulnerability closed?   Edit: The selected answer specifically addresses the question of how to identify if a given CVE has been fixed or not, "Does Ubuntu generally post timely security updates?"  is certainly related but not identical How to convert from one grading scheme to another? I'm trying to figure out the equivalence between grade schemes in different countries.  According to some Google results:   If you are a U.S. citizen, you need a GPA of 3.2+ (it varies, some say 3.2 and some others say 3.6, etc) to apply for a position in the UK (job or university) where the prerequisite is to have a minimum grade of 2:1. In UK 2:1 is earned if you have a 60-69% of the points. I've read that 2:1 is the same as 67%. In Spain we use a grading scheme of 10 points. So according to (2), for have a UK 2:1 you need to have 6-6.9 points. If you are Spanish and have 6.7, when converting it to GPA the result is 1.5.   However, if you take the above points and treat them as a math equation, the final result will be GPA 1.5 = GPA 3.2, which doesn't make any sense.  GPA 3.2 = UK 2:1 UK 2:1 = Spain 6.7 Spain 6.7 = GPA 1.5   Replacing now...  UK 2:1 = GPA 1.5     GPA 3.2 = GPA 1.5   My question is if someone has a better explanation on converting grades obtained in different countries.