Math 411 Homework Assignment 10 Due 4/21/2017 by the end of lecture p p p 1. Show that the subset 2, 2 + i, p3 i} of C is linearly dependent over R. (Hint: Consider a p { p linear combination a 2 + b( 2 + i) + c( 3 i) = 0 and use the fact that 1, i are linearly independent over R to find suitable a, b, c 2 R.) p 2. (a) Prove that the subset {1, 2} of R is linearly independent over Q. (You may use without proof p the fact that 2 is irrational.) p p (b) p Show that 3 is not a linear combination of 1 and 2 with coefficients in Q and conclude that {1, 2} does not span R over Q. 3. Suppose V is a vector space over a field F . Show that a set {v1 , . . . , vn } is a basis of V over F if and only if every element of V can be written uniquely as a linear combination of v1 , . . . , vn , i.e. if w = c1 v1 + + cn vn and w = d1 v1 + + dn vn , then ci = di for i = 1, . . . , n. 4. Verify that the given element is algebraic over Q by writing down an explicit polynomial f (x) 2 Q[x] that has the element as a root. Show your work. (a) 3 + 5i p (b) 1 + 3 2 5. Suppose K is a field extension of F and u 2 K. Show that F (u) = F (au + b) for any a 2 F and b 2 F. (Hint: Recall that F (u) can be viewed as rational functions in u.) 6. Using the various tests for irreducibility discussed in lecture, show that the given polynomials polynomials are irreducible over Q. (a) 10x7 (b) x4 6x4 + 15x2 + 18x 2x2 6 + 8x + 1 7. Suppose K is a field extension of F . (a) Show that [K : F ] = 1 if and only if K = F. (b) Suppose [K : F ] = p for a prime p. Show that for any u 2 K, we have either F (u) = F or F (u) = K. 8. Suppose K is a field extension of F and and u 2 K is algebraic over F . Show that if the minimal polynomial p(x) 2 F [x] of u has odd degree, then F (u) = F (u2 ). (Hint: Consider the chain of extensions F F (u2 ) F (u) and observe that [F (u) : F (u2 )] 2.)