MathDB

Problem 4

Part of 2011 VJIMC

Problems(2)

syseq in group, 3 eqns/variables

Source: VJIMC 2011 1.4

6/1/2021
Let a,b,ca,b,c be elements of finite order in some group. Prove that if a−1ba=b2a^{-1}ba=b^2, b−2cb2=c2b^{-2}cb^2=c^2, and c−3ac3=a2c^{-3}ac^3=a^2 then a=b=c=ea=b=c=e, where ee is the unit element.
group theory
nice FE, irreducibility map

Source: VJIMC 2011 2.4

5/28/2021
Find all Q\mathbb Q-linear maps Φ:Q[x]→Q[x]\Phi:\mathbb Q[x]\to\mathbb Q[x] such that for any irreducible polynomial p∈Q[x]p\in\mathbb Q[x] the polynomial Φ(p)\Phi(p) is also irreducible.
fefunctional equationPolynomialsirreducibilityalgebra