Polynomials

Suppose the nonzero complex number $\alpha$ is a root of a polynomial of degree $n$ with rational coefficients. Prove that $1/\alpha$ is also a root of a polynomial of degree $n$ with rational coefficients.

By the Fundamental Theorem of Algebra, the polynomial $x^3+2x^2+7x+1$ has three complex roots, $\alpha_1, \alpha_2$, and $\alpha_3$. Compute $\alpha_1^3+\alpha_2^3+\alpha_3^3$.

Let $\zeta = e^{\frac{2\pi i}{7}}$ be a primitive 7th root of unity. Find a cubic polynomial with integer coefficients having $\alpha = \zeta + \zeta^{-1}$ as a root.

Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational.

Let $P(z)$ be a polynomial of degree $< k$ with complex coefficients. Let $\omega_1, \ldots , \omega_k$ be the $k$-th roots of unity in $\mathbb {C}$. Prove that

Let $f(x) \in \mathbb {Q}^{x}$ be a polynomial with rational coefficients. Show that there is a $g(x) \in \mathbb {Q}^{x}$, $g \neq 0$, such that $f(x)g(x) = a_2 x^2 + a_3 x^3 + a_5 x^5 + \cdots + a_p x^p$ is a polynomial in which only prime exponents appear.

Let $I$ be the ideal in the ring $\mathbb {Z}^{x}$ generated by $x-7$ and 15. Prove that the quotient ring $\mathbb {Z}^{x}/I$ is isomorphic to $\mathbb {Z}^{15}$.

Let $I$ denote the ideal in $\mathbb {Z}^{x}$, the ring of polynomials with coefficients in $\mathbb {Z}$, generated by $x^3+x+1$ and 5. Is $I$ a prime ideal?

In the ring $\mathbb {Z}^{x}$ of polynomials in one variable over the integers, show that the ideal $I$ generated by 5 and $x^2+2$ is a maximal ideal.

Let $\mathbb {Z}^{n}$ denote the ring of integers modulo $n$. Let $\mathbb {Z}^{n}[x]$ be the ring of polynomials with coefficients in $\mathbb {Z}^{n}$. Let $I$ denote the ideal in $\mathbb {Z}^{n}[x]$ generated by $x^2+x+1$.

Let $\mathbb {Z}$ be the ring of integers, $p$ a prime, and $\mathbf{F}_p = \mathbb {Z}/p\mathbb {Z}$ the field of $p$ elements. Let $x$ be an indeterminate, and set $R_1 = \mathbf{F}_p[x]/ \left\langle x^2-2\right\rangle$, $R_2 = \mathbf{F}_p[x]/ \left\langle x^2-3\right\rangle$. Determine whether the rings $R_1$ and $R_2$ are isomorphic in each of the cases $p = 2, 5, 11$.

Consider the polynomial ring $\mathbb {Z}^{x}$ and the ideal $I$ generated by 7 and $x-3$.

Let $\mathbb {Z}^{x}$ be the ring of polynomials in the indeterminate $x$ with coefficients in the ring $\mathbb {Z}$ of integers. Let $I \subset \mathbb {Z}^{x}$ be the ideal generated by 13 and $x-4$. Find an integer $m$ such that $0 \leq m \leq 12$ and

Give an example, with proof, of a nonconstant irreducible polynomial $f(x)$ over $\mathbb {Q}$ with the property that $f(x)$ does not factor into linear factors over the field $K = \mathbb {Q}^{x}/(f(x))$.

Find all automorphisms of $\mathbb {Z}^{x}$, the ring of polynomials over $\mathbb {Z}$.

Let $R$ denote the ring of polynomials over a field $F$. Let $p_1, \ldots , p_n$ be elements of $R$. Prove that the greatest common divisor of $p_1, \ldots , p_n$ is 1 if and only if there is an $n \times n$ matrix over $R$ of determinant 1 whose first row is $(p_1, \ldots , p_n)$.

Let $f(x)$ be a polynomial over $\mathbb {Z}^{p}$, the field of integers mod $p$. Let $g(x) = x^p - x$. Show that the greatest common divisor of $f(x)$ and $g(x)$ is the product of the distinct linear factors of $f(x)$.

Let $F$ be a subfield of a field $K$. Let $p$ and $q$ be polynomials over $F$. Prove that their greatest common divisor in the ring of polynomials over $F$ is the same as their gcd in the ring of polynomials over $K$.

Show that $x^{10}+x^9+x^8+\cdots +x+1$ is irreducible over $\mathbb {Q}$. How about $x^{11}+x^{10}+\cdots +x+1$?

Let $\mathbb {Z}$ be the ring of integers and $\mathbb {Z}^{x}$ the polynomial ring over $\mathbb {Z}$. Show that

is irreducible in $\mathbb {Z}^{x}$.

Prove that the polynomial $x^4+x+1$ is irreducible over $\mathbb {Q}$.

Prove that if $p$ is a prime number, then the polynomial

is irreducible in $\mathbb {Q}^{x}$.

Prove that $f(x) = x^4+x^3+x^2+6x+1$ is irreducible over $\mathbb {Q}$.

Prove that the polynomial $f(x) = 16x^5 - 125x^4 + 50x^3 - 100x^2 + 75x + 25$ is irreducible over the rationals.

Let $\mathbb {Z}^{3}$ be the field of integers mod 3 and $\mathbb {Z}^{3}[x]$ the corresponding polynomial ring. Decompose $x^3+x+2$ into irreducible factors in $\mathbb {Z}^{3}[x]$.

Factor $x^4+x^3+x+3$ completely in $\mathbb {Z}^{5}[x]$.

Is $x^4+1$ irreducible over the field of real numbers? The field of rational numbers? A field with 16 elements?

Decompose $x^4-4$ and $x^3-2$ into irreducibles over $\mathbb {R}$, over $\mathbb {Z}$, and over $\mathbb {Z}^{3}$ (the integers modulo 3).

Let $a$ be an element in a field $F$ and let $p$ be a prime. Assume $a$ is not a $p$-th power. Show that the polynomial $x^p-a$ is irreducible in $F[x]$.

Let $p$ be a prime integer, $p \equiv 3 \pmod{4}$, and let $\mathbf{F}_p = \mathbb {Z}/p\mathbb {Z}$. If $x^4+1$ factors into a product $g(x)h(x)$ of two quadratic polynomials in $\mathbf{F}_p[x]$, prove that $g(x)$ and $h(x)$ are both irreducible over $\mathbf{F}_p$.

Let $n$ be a positive integer and let $f$ be a polynomial in $\mathbb {R}^{x}$ of degree $n$. Prove that there are real numbers $a_0, a_1, \ldots , a_n$, not all equal to zero, such that the polynomial

is divisible by $f$.

Let $F$ be a field, $F[x]$ the polynomial ring in one variable over $F$, and $R$ a subring of $F[x]$ with $F \subset R$. Prove that there exists a finite set $\left\{ f_1, f_2, \ldots , f_n\right\}$ of elements of $F[x]$ such that $R = F[f_1, f_2, \ldots , f_n]$.

Let $F$ be a finite field with $q$ elements and let $x$ be an indeterminate. For $f$ a polynomial in $F[x]$, let $\varphi_f$ denote the corresponding function of $F$ into $F$, defined by $\varphi_f(a) = f(a)$, ($a \in F$). Prove that if $\varphi$ is any function of $F$ into $F$, then there is an $f$ in $F[x]$ such that $\varphi = \varphi_f$. Prove that $f$ is uniquely determined by $\varphi$ to within addition of a multiple of $x^q - x$.