We say that $[a]$ is a square root of $[b]$ in $\mathbb{Z}/p\mathbb{Z}$ if $[a]^2= [b]$. Prove that if $p$ is prime, then an element of $\mathbb{Z}/p \mathbb{Z}$ has at most two square roots?
Can someone provide a proof that every quadratic residue, when working in $\mathbb Z_p$, where $p$ is a prime, has exactly two roots? Indeed, there cannot be only one root as for any $a^2$, we know $a$ and $-a$ are both roots. So then we need only show that there cannot be more than two- does this follow from some argument for the number of roots a quadratic can have modulo a prime? I'm not well-acquainted with the theorems pertaining to prime moduli.
I read news on the BBC (on its website) this morning. When I was reading the news I noticed the definite article the was missing in a sentence. As per my opinion, the reporter missed the article the. But I also thought that the reporter also had some point as he did not use the article there. So I just wanted to confirm so landed here to get some help on it.     On the website:      It also proclaimed the group's leader, XYZ  (name), as caliph and   "leader for Muslims everywhere".      As per my opinion it should have been:      It also proclaimed the group's leader, XYZ  (name), as the caliph and   "leader for Muslims everywhere".   Source link: single query to print all the rows whose count is greater than 10 I tried to manipulate a cube with the Boolean modifier. I tried to cut out an edge of the cube with a cylinder. Also, the shading result looks awful in FLAT SHADING mode, so I turned on SMOOTH SHADING.   Has anyone an idea how to achieve a better result. (I also cut out a Sphere first, this made the result less awful).   I used Blender Internal and rendered with smooth shading for smoothness.  Using flat shading looks okay, except the "round" cutouts :/  Face normals are okay.  First image is smooth shading.    EDIT: User Pisurquatre suggested the EdgeSplit modifier, works perfect to me (Maybe a little tweaking here and there): Can an ideal in a commutative integral domain be its own square? If $I$ is a non-zero proper ideal of a commutative integral domain, is it possible for $I$ to be its own square? I am learning P&amp;S's Introduction of quantum field theory. My teacher said that if the mass of neutrino is exactly 0, then we should not observe any right-handed neutrinos and left-handed anti-neutrinos according to Weyl's theory. It is because $$ \left( i\gamma^\mu \partial_\mu - m \right) \psi = \left( \begin{array}{cc} -m &amp;  i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right) \\ i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right) &amp; -m \end{array} \right) \left( \begin{array}{c} \psi_L \\ \psi_R \end{array} \right) =0. $$  If we set m=0, then we will have:  \begin{align} i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right) \psi_L &amp;= 0; \\ i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right) \psi_R &amp;= 0. \end{align}  And if these formulas can describe neutrinos we must conclude that there are two types of neutrinos.  My confusions are:   It is known that the mass of neutrino is not 0, therefore there are chances (depending on the mass) to find right-handed neutrinos and left-handed anti-neutrinos. But we don't. If I accept that there are only these two types of neutrinos (right-handed anti-neutrinos and left-handed neutrinos), this property of neutrino causes the parity violation, so parity violation must relate to angular momentum and therefore spinor space. Am I write? Are there any reasons why neutrino is so weird that is has mass but still has helicity + Or - 1. This goes against with what I have learned that only when a particle  moves at the speed of light can its spin aligned with its momentum. $\dim(U_1+U_2) = \dim U_1 +\dim U_2 - \dim(U_1\cap U_2).$  I want to make sure that my intuition is correct. Suppose we have two planes $U_1,U_2$ though the origin in $\mathbb{R^3}$. Since the planes meet at the origin, they also intersect, which in this case is a one-dimensional line in $\mathbb{R^3}$. To obtain the dimension of $U_1$ and $U_2$, we add the dimensions of the planes (4), and the subtract the dimensions of the line (1), which results in (3).   *additional question(s):   Can we generalize this notion to $\mathbb{F^{n}}$?  Suppose we have an additional case where $U_1$ and $U_2$ are planes in $\mathbb{R^3}$, but $U_1 \subseteq U_2$. In this instance, $dim(U_1 + U_2) &lt; 3$, because the first two-dimensional plane is contained in the second and as a result, the dimensions of the subspaces when summed cannot exceed two. Since both subspaces $U_1,U_2$ are two dimensional and $U_1 \subseteq U_2$, then their intersection is also two-dimensional, concluding $dim(U_1+U_2)=2+2-2 = 2$.  Is this proper intuition? What is the sum of non-real roots of the polynomial equation? Where $\phi(n)$ is the Euler phi function, how do you find all $n$ such that $\phi(n)|n$? N circles in the plane External hard drive and Recycle Bin behavior When my Seagate external hard drive is plugged in, its $RECYCLE.BIN folder automatically contains all the files contained in the Recycle Bin of the PC. Is it true that actually we don't have any Recycle Bin of external hard drive, rather it is the Recycle Bin of the PC? Why is it so? No input file specified Why "a user" instead of "an user"? I am getting confused while using an article an. Because an is used while referring to vowels (a,e,i,o,u).  But today I read an article where for user they used a user.  My question is, why was an not used?  Could someone explain this with an example? The $n$ complex $n$th roots of a complex number $z$ Find the MLE of the unknown parameter $\theta$ when $X_1,X_2,...,X_n$ is a sample from the distribution whose density function is: $$f_X(x) = \frac12e^{-|x-\theta|}, -\infty&lt;x&lt;\infty$$ What I did so far: $$\text{maximize }(\frac12)^ne^{-|x_1-\theta|-|x_2-\theta|-...-|x_n-\theta|} \\ \text{thus we want to minimize }|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|$$ Since $|x_i-\theta|$ is positive for all $i$, we want to solve for $\theta$ so: $$|x_1-\theta|+|x_2-\theta|+...+|x_n-\theta|=0$$ What do I do from here? I remember I saw a apt mirror somewhere containing older distributions but I cannot find it anymore. Does anybody know where to find it? Show that $n-m$ is a multiple of 9 when $n$ and $m$ have same digits In Windows 7, How to query times, when the computer was locked? Is there a way to get a list of the following times:  whenever windows is locked (using windows+l combo)  whenever windows is unlocked in the past week?   I've tried poking in the event log, to no avail. Many thanks! "Open shaming of anyone..." Open shaming of anyone who spoke out discouraged responsible dissent.   How does open with shaming work, grammatically and meaningfully?   Is discouraged modifying "responsible dissent? Wordpress speed slow after .5 million single image post via wp-cli If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here?