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This is a slightly abridged preliminary version of the book
Mathematics Via Problems: from olympiades
and math circles to profession. Part I. Algebra.
A. Skopenkov. 2021, AMS, Providence.
The page of the book at the AMS Bookstore:
https://bookstore.ams.org/cdn-1613114724929/mcl-25/.
PartsII(Geometry)andIII(Combinatorics)editedbyM.Skopenkov
and A. Zaslavsky are to appear.
The preliminary version is put at
http://www.mccme.ru/circles/oim/algebra_eng.pdfwithakind
permission of the AMS. For the Russian version see
https://biblio.mccme.ru/node/6017 and (the online part)
https://www.mccme.ru/circles/oim/materials/sturm.pdf.
From the introduction:
Adeep understanding of mathematics is useful both for mathe-
maticians and for high-tech professionals. In particular, the ‘profes-
sion’ in the title of this book does not necessarily mean the profession
of mathematics.
This book is intended for high school students and undergradu-
ates (in particular, those interested in Olympiads)...
The book can be used both for self-study and for teaching. This
book attempts to build a bridge (by showing that there is no gap)
between ordinary high school exercises and the more sophisticated,
intricate and abstract concepts in mathematics. The focus is on
engaging a wide audience of students to think creatively in applying
techniques and strategies to problems motivated by ‘real world or
real work’...
Much of this book is accessible to high school students with a
strong interest in mathematics...
We ascribe to the tradition of studying mathematics by solving
and discussing problems. These problems are selected so that in the
process of solving them the reader (more precisely, the solver) mas-
ters the fundamentals of important ideas, both classical and modern.
Themainideasaredeveloped incrementally with olympiad-style ex-
amples; in other words, by the simplest special cases, free from tech-
nical details. In this way, we show how you can explore and discover
these ideas on your own.
1
Alist of corrections
αn αm
P. 13. At the beginning of problem 1.6.3 replace p by p .
n m
P. 13. Add after problem 1.6.6: ‘Problem 1.6.6.c is complicated [An90]. (I
amgrateful to N. Osipov for bringing this to my attention.)’
P. 108. Add ‘E. Kogan’ to the acknowledgements.
P. 111. In Remark 8.1.9 replace
‘the words “one of the roots” replaced by “none of the roots”’ by
‘the words “one of the roots is not expressible” replaced by “none of the
roots is expressible”.
P. 114. Add to the end of the third paragraph from below.
Another exposition of the proof of the Kronecker theorem 8.1.14 is given in
[PC19]. That exposition is similar to §8.4.G but is unnecessarily complicated.
E.g. it uses both the dimension argument as in Lemma 8.4.21 and Gauss’ degree
lowering Theorem 8.1.15. In fact, Theorem 8.1.15 alone is sufficient for a short
proof, see §8.4.G.
P. 114. Add between the third and the fourth paragraphs from below.
For earlier versions of this section see [Sk08, Skod].
P. 121. In 8.2.12.c replace ‘Prove (b)’ by ‘Prove the analogue of (b) for ε7
replaced by cos(2π/7)’.
P. 145. Replace Remark 8.3.36 by the following.
In solution of Problem 8.2.3c we constructed polynomials
f (x,y,z), f (x,y,z), f (x,y,z),
1 2 3
p (u,v,w), p (u,v,w,t ), p (u,v,w,t ,t ), p (u,v,w,t ,t ,t )
0 1 1 2 1 2 3 1 2 3
with complex coefficients such that
2
f =p (σ ,σ ,σ )
1 0 1 2 3
3
f =p (σ ,σ ,σ ,f )
2 1 1 2 3 1 .
3
f =p (σ ,σ ,σ ,f )
3 2 1 2 3 1
x=p (σ ,σ ,σ ,f ,f ,f )
3 1 2 3 1 2 3
(If we allow p to be a rational function, then we can omit the third equation
3
and f3 in the fourth equation.) There are no polynomials with real coefficients
such that the above system holds.
P. 158. In Lemma 8.4.16(a) replace ‘Suppose that’ by ‘Suppose that F = F,’.
P. 159. In the last two paragraphs replace twice ‘Linear Independence
lemma’ by ‘Linear Independence lemma 8.4.14.b and F = F,’.
P. 161. Proof of Lemma 8.4.19(a) contains a gap. ‘We repeat the first three
paragraphs’ cannot be done because irreducibility of H(x,r) is given over F[r]
but is required over F[r,ε]. Perhaps the other proof from the Russian original
omitted in the English translation does work.
P. 163-166. Add references
2
[An90] W.S. Anglin. The Square Pyramid Puzzle, Amer. Math. Monthly,
97 (1990) 120–124.
[PC19]Y.PanandY.Chen. OnKronecker’sSolvabilityTheorem,arXiv:1912.07489.
[Sk08] A. Skopenkov. Some more proofs from the Book: solvability and
insolvability of equations in radicals, arXiv:0804.4357.
3
❈♦♥t❡♥ts
❋♦r❡✇♦r❞ ✈✐✐
Pr♦❜❧❡♠s✱ ❡①❡r❝✐s❡s✱ ❝✐r❝❧❡s✱ ❛♥❞ ♦❧②♠♣✐❛❞s ✈✐✐
❲❤②t❤✐s ❜♦♦❦✱ ❛♥❞ ❤♦✇ t♦ ✉s❡ ✐t ✈✐✐✐
❊♥❣❧✐s❤✲❧❛♥❣✉❛❣❡ r❡❢❡r❡♥❝❡s ✈✐✐✐
■♥tr♦❞✉❝t✐♦♥ ①✐
❲❤❛t t❤✐s ❜♦♦❦ ✐s ❛❜♦✉t ❛♥❞ ✇❤♦ ✐t ✐s ❢♦r ①✐
▲❡❛r♥✐♥❣ ❜② ❞♦✐♥❣ ♣r♦❜❧❡♠s ①✐✐
❆♠❡ss❛❣❡✳ ❇② ❆✳❨❛✳❑❛♥❡❧✕❇❡❧♦✈ ①✐✐
❖❧②♠♣✐❛❞s ❛♥❞ ♠❛t❤❡♠❛t✐❝s ①✐✐✐
❘❡s❡❛r❝❤ ♣r♦❜❧❡♠s ❢♦r ❤✐❣❤✲s❝❤♦♦❧ st✉❞❡♥ts ①✐✐✐
❍♦✇ t❤✐s ❜♦♦❦ ✐s ♦r❣❛♥✐③❡❞ ①✐✐✐
❙♦✉r❝❡s ❛♥❞ ❧✐t❡r❛t✉r❡ ①✐✈
❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ①✐✈
●r❛♥t s✉♣♣♦rt ①✈
◆✉♠❜❡r✐♥❣ ❛♥❞ ♥♦t❛t✐♦♥ ①✈
◆♦t❛t✐♦♥ ①✈
P❛rt ✶✳ ◆✉♠❜❡r ❚❤❡♦r②✱ ❆❧❣❡❜r❛✱ ❛♥❞ ❈❛❧❝✉❧✉s ✶
❈❤❛♣t❡r ✶✳ ❉✐✈✐s✐❜✐❧✐t② ✸
✶✳ ❉✐✈✐s✐❜✐❧✐t② ✭✶✮ ✸
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✹
✷✳ Pr✐♠❡ ♥✉♠❜❡rs ✭✶✮ ✺
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✻
✸✳ ●r❡❛t❡st ❈♦♠♠♦♥ ❉✐✈✐s♦r ✭●❈❉✮ ❛♥❞ ▲❡❛st ❈♦♠♠♦♥ ▼✉❧t✐♣❧❡ ✭▲❈▼✮
✭✶✮ ✼
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✽
✹✳ ❉✐✈✐s✐♦♥ ✇✐t❤ r❡♠❛✐♥❞❡r ❛♥❞ ❝♦♥❣r✉❡♥❝❡s ✭✶✮ ✾
❍✐♥ts ✶✵
✺✳ ▲✐♥❡❛r ❉✐♦♣❤❛♥t✐♥❡ ❡q✉❛t✐♦♥s ✭✷✮ ✶✵
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✶✶
✻✳ ❈❛♥♦♥✐❝❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✷✯✮ ✶✷
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✶✸
✼✳ ■♥t❡❣❡r ♣♦✐♥ts ✉♥❞❡r ❛ ❧✐♥❡ ✭✷✯✮ ✶✹
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✶✺
❈❤❛♣t❡r ✷✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♠♦❞✉❧♦ ♣ ✶✼
✶✳ ❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠ ✭✷✮ ✶✼
❙✉❣❣❡st✐♦♥s✱ s♦❧✉t✐♦♥s ❛♥❞ ❛♥s✇❡rs ✶✽
✐
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