Question

In: Advanced Math

Show that, for each natural number n, (x − 1)(x − 2)(x − 3). . .(x...

Show that, for each natural number n, (x − 1)(x − 2)(x − 3). . .(x − n) − 1 is irreducible over Q, the rationals.

Solutions

Expert Solution

We show the irreducibility of the polynomial

over by using the method of contradiction.

Note that the polynomial and hence is irreducible over if and only if it is irreducible over Now, let if possible, there exist monic integer coefficient polynomials of degrees such that

Then,

. Since the polynomials are integer coefficient polynomials and hence

Now, the polynomial

is a polynomial of degre (since degrees of are and the degree of is with ) and

Hence, is a polynomial of degree at most but it has roots, namely, which is possible only when the polynomial is identically which implies that which is a contradiction (since are monic integer coefficient polynomials).

Hence it follows that the polynomial is irreducible over

Colby Messinger answered 1 year ago

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