HINTS

1. Try to express sums and products of elements of the set in the given form.
2. Can you express a² in the given form? Use that a is not a rational number and that it is a zero of X³ - 2.

Verify the properties of a subring.

To determine the inverse, solve axa^-1 = y.

1. In the various requirements for a ring you will have to show equality of two functions. Such a verification comes down to checking that the two functions agree at every point. It is there that you can make use of the fact that R itself is a ring.
2. Show injectivity and surjectivity and/or find the inverse map. Use that a subset A of S can be recovered from f_A as f_A^-1(1) (= {s S | f_A(s) = 1}.
3. Find a relation between multiplication and intersection of sets. Likewise, find a relation between addition and union of sets.

Just check the definition of morphism.

1. Use the definition.
2. Show that Ker(g o f) is contained in Ker(f) and vice versa.
3. Show that Ker(g) is contained in f(Ker(g o f)) and vice versa.

To determine the kernel first find a suitable degree one polynomial p in the kernel such that the quotient of any polynomial in the kernel by p has integer coefficients.

To find the inverse, try replacing X by cX + d for suitable c and d.

If ab = 0 and , then ab = a0.

Consider the subring generated by the elements a₁/b₁, ..., a_n/b_n and assume that a_i and b_i are relatively prime for i = 1, 2, ..., n. What prime numbers occur in the denominators of elements of the subring? (Assuming that we represent them as quotients of relatively prime integers.)

Verify that a subset L of a field K is a subfield of K if it satisfies:

• 0, 1 L, and
• x, y L implies x-y L, and
• x, y L and y 0 imply x/y L.

From (a + bd) (a - bd) = a² - b²d construct an inverse of a + bd in Q + Qd.

The map is well defined.
Show that a/b = c/d implies ab^-1 = cd^-1.

Injectivity.
Use a Theorem and compute the kernel of F. What conclusion do you draw from ab^-1 = 0?

The image is the smallest subfield of K containing R.
First show that the image of F is a subfield of K; then show that any subfield L of K containing R contains F(R).

Clearly (1,1) eaq (2,2) and (2,2) eaq (0,2).

1. Use the theorem on the ideal generated by a subset V;
2. First show: 1 (V) implies (V) = R;
3. Use the theorem on the ideal generated by a subset V;
4. f(X) = 2g(X) + Xh(X) is a polynomial with even constant term f(0).

Use X = ((X + Y) + (X - Y))/2.

1. Consider (2 + 2i)i.
2. Verify the conditions in the definition of an ideal.

1. If all coefficients of fg (the product of two polynomials) are divisible by p, then either all coefficients of f or all coefficients of g (or both) are divisible by p. Why?
2. 5 is divisible by 1 + 2i.
3. Suppose f is not in the ideal and we add it to the ideal. Argue, using polynomial division and division of the coefficients by 2, that 1 is in the ideal.

Use the definition of Cartesian product to check the definition of ideal.

The map : Z/2Z[X] -> Z/2Z[X] defined by (f(X)) = f(X + 1) is easily seen to be an automorphism of the ring Z/2Z[X]. It maps the ideal (X³ + X + 1) onto the ideal ((X + 1)³ + (X + 1) + 1) = (X³ + X² + 1).

1. The only irreducible polynomial of degree two in Z/2Z[X] is X² + X + 1 but (X² + X + 1)² = X⁴ + X² + 1 f.
2. Reduce the elements modulo f. This yields the a + b + c² + d³ representation of the elements.
3. Use Lagrange's Theorem to find the order of in the multiplicative group of S.

   For each i, i=0, 1, 2, ..., 14, compute the a + b + c² + d³ representation of ⁱ (this is the so-called logarithm table of the field).

4. The nonzero elements of a subfield of order 4 have the property that their third power equals 1. There are only three such elements: ⁰, ⁵ and ¹⁰.

1. Use the definition.
2. What does the kernel have to do with the polynomial (X - 1)X(X + 1)?
3. Describe the set as a residue class.

How many a's and b's are there? How many irreducible polynomials with leading coefficient 1 are there? Alternatively, given an irreducible polynomial of degree 2 with leading coefficient 1, say, complete the square to transform to the polynomial X² - 2 or X² + 2.

First show that -1 = 1 and then evaluate (x + y)².