Q12E ... [FREE SOLUTION]

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Chapter 4: Q12E (page 119)

Short Answer

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$I t i s p r o v e d t h a t f (x) i s i r r e d u c i b l e i n F [x] .$

Step by step solution

Step-by-Step Solution Step 1: Consider

To prove the result, we need to prove the contrapositive of the result.

Prove the result

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Most popular questions from this chapter

Use mathematical induction to prove Corollary 4.17.

(a) Let $f (x) . g (x) \in F [x]$ . If role="math" localid="1648080019147" $f (x) | g (x)$ and $g (x) | f (x)$ , show that $f (x) = c g (x)$ for some non-zero $c \in F$ .

(b) If $f (x)$ and $g (x)$ in part (a) are monic, show that $f (x) = g (x)$ .

Which of the following subsets of are sub rings of ? Justify your answer:

All polynomials with constant term $0_{R}$ .
All polynomials of degree 2.
All polynomials of degrees $\leq k$ , where k is a fixed positive integer.
All polynomials in which the odd powers of x have zero coefficients.
All polynomials in which the even powers of x have zero coefficients.

Question:Determine if $h (x)$ is a factor of $f (x)$ .

(a) $h (x) = x + 2 and f (x) = x^{3} - 3 x^{2} - 4 x - 12 in ℝ [x]$

(b) $h (x) = x - \frac{1}{2} and f (x) = 2 x^{4} + x^{3} + x - \frac{3}{4} in ℚ [x]$

(d) $h (x) = x - 3 and f (x) = x^{6} - x^{3} + x - 5 in ℤ_{7} [x]$

Show that there are infinitely many integers $k$ such that

x^{9} + 12 x^{5} - 21 x + k

is irreducible in

ℚ [x]

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Short Answer

Step by step solution

Step-by-Step Solution Step 1: Consider

Prove the result

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