Kronecker's Theorem: An Example

In a previous blog, I presented Kronecker's Theorem.

Today, I will show how it can be used to prove that an equation is not algebraically soluble.

The content in today's blog is taken directly from David Antin's translation of Heinrich Dorrie's 100 Great Problems of Elementary Mathematics.

Example: x⁵ - ax - b = 0

Let's assume the following:

(1) a,b are positive integers divisible by a prime p

(2) b is not divisible by p²

(3) 4⁴a⁵ is greater than 5⁵b⁴

Here's the analysis:

(1) Using Eisenstein's Criteria, the equation is irreducible over the set of rational numbers. [see Theorem 1, here].

(2) From Sturm's Theorem [see Theorem, here], it is clear that it possesses three real roots and two complex roots since:

(a) We build the following Sturm Chain (see here for details on Sturm Chains):

P₀ = x⁵ - ax - b [The equation itself]

P₁ = 5x⁴ - a [The first derivative, see here for review if needed]

P₂ = 4ax + 5b [The remainder from P₀ and P₁, see here for view if needed]

P₃ = 4⁴a⁵ - 5⁵b⁴ [The remainder from P₁ and P₂]

(b) We know that there are 5 roots from the Fundamental Theorem of Algebra [see here for proof]

(c) From an analysis, I did earlier (see Example 2, here), we know that there are three real roots.

(3) Assume that the equation is algebraically soluble.

(4) Then, from Kronecker's Theorem [see Theorem 4, here], it either has only one real root or all real roots.

(5) But this is not the case from Sturm's Theorem so we have a contradiction.

(6) Thereofore, we reject our assumption in step #3 and conclude that the equation is not algebraically soluble.

References

• Heinrich Dorrie (Translated by David Antin), 100 Great Problems of Elementary Mathematics (Dover, 1965)