MHB Is a^2 - 6a + 12 Irreducible Over the Integers?

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The discussion centers on the polynomial a^2 - 6a + 12, which is stated to be irreducible over the integers, meaning it cannot be factored into polynomials with integer coefficients. The expression is derived from the factorization of 8 + (a - 2)^3. Participants seek clarification on the author's claim and request additional examples of irreducible polynomials. The conversation emphasizes understanding irreducibility in the context of polynomial factorization. Overall, the focus is on the properties of the polynomial and the concept of irreducibility in algebra.
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In the textbook, the author showed that 8 + (a - 2)^3 factors out to be a(a^2 - 6a + 12).

The author goes on to say "...the expression a^2 - 6a + 12 is irreducible over the integers."

What does the author means by the statement?
 
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It simply means that the polynomial $a^2-6a+12$ cannot be factored into the product of two polynomials with coefficients that are integers. :D
 
Can you provide me with a few more irreducible examples?
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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