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Algebra
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20 Nov 2019

Decide which of the following polynomials is irreducible:

x7+11x3-33x+22 over Q;

x3-7x2+3x+3 over Q;

x4+1 over Q;

xq+1 over Q for any q=2n;

1+x+x2 over Q;

1+x+x2+...+xp-1 over Q for any prime p.

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Elin Hessel
Elin HesselLv2
27 Jan 2019
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1. Let V be an inner product space. Prove the Pythagoreantheorem: if v and w E V are orthogonal then ||v+w||^2 =||v||^2 + ||w||^2. (Suggestion: start with ||v+w||^2 = (v+w,v+w).)


2. Let V=P2, the polynomials of degree less than/equal to 2 withcoefficients in R, and let (.,.): VxV -> R be the map

(p,q) = p(0)q(0) + p(1)q(1) + p(2)q(2) for any two polynomialsp(x),q(x) E V. So, for example,

(x+2, x^2-1) = (0+2)(0^2-1) + (1+2)(1^2 - 1) + (2+2)(2^2-1) =10.


a) Show that the function(.,.) defines an inner product onV.


b) Is the set A=(1,x,x^2) an orthogonal set? (with respect tothe inner product above.)


c) Is the set B=(1,x-1,3x^2-6x+1) an orthogonal set?


d) Calculate ||1||^2, and ||3x^2-6x+1||^2


e) Is the set B an orthonormal set?


f) Write f=x^2-6x+12 as a linear combination of the vectors inB.

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