Mathematics About the Number 2017
Happy New Year 2017!!
Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek.
Add to solve later Add to solve later If Eigenvalues of a Matrix $A$ are Less than $1$, then Determinant of $I-A$ is Positive
Problem 237
Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$.
Then show that the determinant \[ \det(I-A) >0,\] where $I$ is the $n \times n$ identity matrix.
Add to solve later Is a Set of All Nilpotent Matrix a Vector Space?
Problem 236
Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer.
Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.
Add to solve later Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues
Problem 235
Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$.
Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$.
(Nagoya University, Linear Algebra Final Exam Problem)
Read solution
Add to solve later Polynomial $x^4-2x-1$ is Irreducible Over the Field of Rational Numbers $\Q$
Problem 234
Show that the polynomial
\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.
Add to solve later Explicit Field Isomorphism of Finite Fields
Problem 233
(a) Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.
(b) Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.
(c) Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.
Add to solve later Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Problem 232
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Add to solve later Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group
Problem 231
Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.
Add to solve later Galois Group of the Polynomial $x^2-2$
Problem 230
Let $\Q$ be the field of rational numbers.
(a) Is the polynomial $f(x)=x^2-2$ separable over $\Q$?
(b) Find the Galois group of $f(x)$ over $\Q$.
Add to solve later Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field
Problem 229
Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.
For any nonzero element $a\in \F_p$, prove that the polynomial
\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.
(Dummit and Foote “Abstract Algebra” Section 13.5 Exercise #5 on p.551)
Add to solve later Characteristic of an Integral Domain is 0 or a Prime Number
Problem 228
Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Add to solve later Subgroup Containing All $p$-Sylow Subgroups of a Group
Problem 227
Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.
Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.
Then show that $N$ contains all $p$-Sylow subgroups of $G$.
Add to solve later If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup
Problem 226
Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.
Add to solve later Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$
Problem 225
Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.
Add to solve later 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$
Problem 224
In the ring
\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\]
show that $5$ is a prime element but $7$ is not a prime element.
Add to solve later A Prime Ideal in the Ring $\Z[\sqrt{10}]$
Problem 223
Consider the ring
\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\]
and its ideal
\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\]
Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.
Add to solve later Dimension of Null Spaces of Similar Matrices are the Same
Problem 222
Suppose that $n\times n$ matrices $A$ and $B$ are similar.
Then show that the nullity of $A$ is equal to the nullity of $B$.
In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of $B$.
Add to solve later Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
Problem 221
Let $p$ be a prime number. Let
\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.
Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.
Add to solve later If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
Problem 220
Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.
Add to solve later Use Lagrange’s Theorem to Prove Fermat’s Little Theorem
Problem 219
Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.
Add to solve later 
