Tagged: irreducible polynomial

Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements

Problem 529

Let $\F_3=\Zmod{3}$ be the finite field of order $3$.
Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$.

(a) Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have?

(b) Let $ax+b+I$ be a nonzero element of the field $\F_3[x]/(x^2+1)$, where $a, b \in \F_3$. Find the inverse of $ax+b+I$.

(c) Recall that the multiplicative group of nonzero elements of a field is a cyclic group.

Confirm that the element $x$ is not a generator of $E^{\times}$, where $E=\F_3[x]/(x^2+1)$ but $x+1$ is a generator.

 
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Differentiating Linear Transformation is Nilpotent

Problem 453

Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]

(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.

(b) Compute $A^3$, where $A$ is the matrix obtained in part (a).

(c) If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?

(d) Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$.
Prove that without any calculation that the matrix $A$ is nilpotent.

 
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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 $\F_3$.

 
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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)

 
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