IEEE - Aerospace and Electronic Systems - July 2022 - 46

News & Information
DOI. No. 10.1109/MAES.2022.3165765
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Giovanni Ansaldo and Algebraic Equations in the XIX Century
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Marco Frasca , MBDA Italia S.p.A., 00131 Roma, Italy
Alfonso Farina
Roma, Italy
Giovanni Ansaldo (see Figure 1) was an Italian entrepreneur,
an engineer, an architect, and an academic. He
started the great tradition of Italian industry that has its
greatest achievements in technology and defense. He was
mostly interested in solving mathematical problems. We
testified to this interest with an article that appeared on the
site of the Foundation taking his name. [1] We briefly
summarize the content of that article here.
In the first decades of the XIX century, there were
some relevant problems on solving algebraic equations.
For sure, there was the Abel-Ruffini theorem, stated by
Paolo Ruffini (1765-1822), and definitively proven by
Niels Abel (1802-1829) in 1824. This theorem states that
algebraic equations of degree greater than four cannot be
solved by simple operations involving at most radicals.
Similarly, Carl Friedrich Gauss (1777-1855) proved in
1799 that the number of solutions of algebraic equations is
equal to the maximum degree of the equation itself (the
fundamental theorem of algebra). Anyway, included in
such solutions there must be also imaginary solutions
involving the imaginary unit i ¼
p
ffiffiffiffiffiffiffi
1
, Leonardo, Consultant BU Land and Naval Radar, 00131
quartic roots of unity. This attempt by Ansaldo is again a
step forward for the great revolution started byEvariste
Galois (1811-1832) with the introduction of the group
theory. It should be said that Giovanni Ansaldo would not
have been aware of Galois' work as this was made widely
known just some years after Galois' untimely death.
Galois, before his death at the age of twenty
Evariste
. A vast part of
mathematics started from these complex numbers, with
essential contributions from Euler and Augustin-Luis
Cauchy (1789-1857). We should not forget also
Joseph-Louis Lagrange (1736-1813; born Giuseppe-Luigi
Lagrangia) who introduced an original approach to solve
third-degree equations using the cubic roots of unity. This
was an approach hinting at group theory, as we will see in
the following.
Starting from Lagrange's idea, Ansaldo, in his " Foglio
di Calcolo " (see Figure 2) applies an identical approach to
fourth-degree equations. In this case, he started from the
Authors' current address: Marco Frasca is with MBDA
Italia S.p.A., 00131 Roma, Italy (e-mail: marco.frasca@
mbda.it); Alfonso Farina is with Leonardo, Consultant
BU Land and Naval Radar, 00131 Roma, Italy (e-mail:
alfonso.farina@outlook.it).
Manuscript received March 28, 2022; accepted April 2,
2022, and ready for publication April 5, 2022.
Review handled by Daniel O'Hagan.
0885-8985/22/$26.00 ß 2022 IEEE
46
in a duel, revolutionized this important area of mathematics,
extending the ideas of his predecessors, by introducing
group theory and showing definitively that no solution can
be obtained in closed form, just using algebraic operations
and consideration of the roots of the equation, for algebraic
equations of degree higher than four. His work
became widely known only after its publication, due to
Joseph Liouville (1809-1882), in 1846. This has been the
start of a wonderful area of mathematics, group theory,
that has its effects till today in many fields of application,
last but not least the motivation to build the large hadron
collider at CERN in Geneva. [3] The theory of groups is
also fundamental in the design of codes for radar waveforms.
[4]-[6]
Figure 1.
Giovanni Ansaldo (Genoa, 1815-Genoa, April 29, 1859).
IEEE A&E SYSTEMS MAGAZINE
JULY 2022
https://orcid.org/0000-0002-0163-0659 https://orcid.org/0000-0003-3247-2427
IEEE - Aerospace and Electronic Systems - July 2022

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