An introduction to the modern theory of equations, by Florian Cajori.

THE ALGEBRAIC SOLUTION OF EQUATIONS 231
Ex. 5. In V, Ex. 1, let d = cl, z = cX, where j/ and X are numbers in
the domain 0(i), or in any other domain. Show that x5 + cx + d 0 is
always metacyclic when
55A4X
((X- 1)4(X2 - 6 x+ 26)
55/5X
(X - 1)4(X2 6 X + 25)
Ex. 6. Construct the metacyclic quintic in which. = v/2, X =6.
See Ex. 5.
Ex. 7. Is 5 + x + 1 =0 metacyclic?
Ex. 8. There is a theorem to the effect that all irreducible, metacyclic
equations of the sixth degree in a domain 2 may be found by adjoining to
2 a square root and then forming in the enlarged domain all cubic equations. See Weber's Algebra, Vol. II, 1896, p. 296. Accordingly, adjoining
v/2 to (2), we may write X3 + x + 1 + \/2 = 0 and obtain, by transposing
V/2 and squaring, the metacyclic sextic x6 + 2 x4 + 2 x3 + x'2-x —=0.
Derive similar equations, using the radical v/3.
Ex. 9. Show that x5 + 5pxt + 10p2X3 + 10p32 + 5 p4X + p5 -1 = 0 is
metacyclic. Also determine its Galois group.
Increase its roots by p.
Ex. 10. Show that y5 + py3 + - p2y + r = 0 is metacyclic.
Take P
J -- 5 z'
Ex. 11. Prove that equation V in Ex. 1 can have no rational root when
c = ~- 1. Then prove that, if xs i x + d = 0 is solvable, it is reducible.
Ex. 12. Show that x5 - A = 0, where A is not a perfect fifth power, is
metacyclic and has the group G20(5) in the domain 1Q(, A).
Ex. 13. Prove that an irreducible equation f(x) =0 of the prime degree
n can become reducible by adjoining a radical V/at, where m is prime, only
when m = n.
Let y7m - a = 0 I
be irreducible, let it have the roots y, wy,..., &m-ly, where w is a complex
rnth root of unity. Let f(x) = 0 become reducible when y is adjoined to
Q, so that f(x)=fi(x, y) f2(x, y), II
the coefficient of the highest power of x in each polynomial being unity.
We may consider I and II as equations in the same domain, having the