Why is x^4+x+1 irreducible over the rationals ℚ? Use the Mod p Irreducibility Test скачать в хорошем качестве

Why is x^4+x+1 irreducible over the rationals ℚ? Use the Mod p Irreducibility Test

Let f(x) = x^4 + x + 1 ∈ ℤ[x]. We claim the quartic (degree 4) f(x) is an irreducible polynomial over the field of rational numbers ℚ. To verify this, we use the Mod p irreducibility test (with p = 2) as well as some computations. First, let f̅(x) be the polynomial in ℤ2[x] obtained from f(x) by taking coefficients mod 2, which has the same formula as f(x) in this case: f̅(x) = x^4 + x + 1 ∈ ℤ2[x]. We verify that f̅(x) has no linear or cubic factors by confirming that f̅(0) ≠ 0 and f̅(1) ≠ 0. But does the quartic f̅(x) have any quadratic factors? We need to list out the possibilities and rule them out. The hardest one has to be ruled out using long division modulo 2. #AbstractAlgebra #PolynomialRing #FieldTheory Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.