# Tag Info

Accepted

### Irreducible polynomial in $\Bbb{Z}_2[x]$

The answer is no. The first counterexample is \begin{multline*} x^{23}+x^{21}+x^{19}+x^{17}+x^{15}+x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+1 \\= \left(x^3+x^2+1\right) \left(x^4+x+1\right) \left(x^{16}+x^{...
• 79k

### Using Eisenstein's Criterion with a transformation

Yes; this works because polynomials (with non-zero constant term) in $R[x]$ are reducible if and only if they are reducible in $R[x,x^{-1}]$. Related (maybe duplicate): Substituting $y=1/x$ to apply ...
• 927
Accepted

### Prove that $x^4+3x^3+3x^2-5$ is irreducible over $\mathbb{Q}$

First of all, by Gauss's lemma, irreducibility over $\mathbb{Q}$ is here equivalent to irreducibility over $\mathbb{Z}$. There are no rational roots, so the only possibility is that the polynomial is ...
• 40.2k
Accepted

### Irreducibility of $X^4-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})$.

Note that $f(x)=x^8-2$ is irreducible over $\mathbb{Q}$ by Eisenstein. Also, $x^8-2=(x^4-\sqrt{2})(x^4+\sqrt{2})$. So if $\alpha\in\mathbb{C}$ is any root $x^4-\sqrt{2}$ then $f$ is the minimal ...
• 40.2k
Accepted

### Investigate whether the polynomial $q(x) = 2x^5 - 78x^3 + 39x + 21$ is irreducible in $\mathbb{F}_{13}[x]$.

It is a classical result that for any field $K$ and any $P\in K[X]$, we have $$P(a)=0\iff X-a\mid P.$$ In particular, an irreducible polynomial in $K$ of degree $\geqslant 2$ has no roots in $K$. We'...
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### Investigate whether the polynomial $q(x) = 2x^5 - 78x^3 + 39x + 21$ is irreducible in $\mathbb{F}_{13}[x]$.

The claim is that it decomposes into two smaller polynomials neither of which is a unit, as a factor $x - a$ is not a unit in $\mathbf{F}_{13}[x]$. To answer your second question, think about what ...
• 630

• 399k
1 vote
Accepted

### Factorization and irreducibilty for $x^n-2x^m+1$ trinomials.

This is a consequence of known results on irreducibility and factorization of trinomials, e.g. see Schinzel's theorem below, excerpted from this 2020 survey by Koley and Reddy. Beware there is typo ...
• 272k
1 vote

1 vote

### Is a polynomial monotone when the first derivative has only imaginary roots?

Consider $f(x):=2x^3-9x^2+12x$. Then $f(x)$ has just one real zero, at $x=0$, the other two being complex. Also, its gradient $f'(x)=6(x^2-3x+2)=6(x-1)(x-2)$ is $12$ at $x=0$ and $x=3$, while it has a ...
• 18.5k
1 vote

### Is a polynomial monotone when the first derivative has only imaginary roots?

Not necessarily. The first derivative being positive at the lower and upper bounds of the range indicates that the function is increasing at those points. However, the function could still have other ...
• 562
1 vote
Accepted

For the sake of contradiction assume $f(x)=g(x)h(x)$ is a non-trivial factorization. Then from $f(b)$ being a prime we can assume (without loss of generality) that $|g(b)|=1$. Now let $g(x)=a\prod (x-\... • 16.6k 1 vote Accepted ### irreducibility in p-adic field I claim that$f$is irreducible, independent of concrete choice of$u$. Note that we know all the roots of$f$: They are of the form$\zeta_p^k u^{1 / p}$for$\zeta_p$a primitive$p\$th root of unity,...
• 2,947

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